fof(t77_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) => k2_card_2(A,A) = A ) ) ), inference(mizar_proof,[status(thm)],[s1_xboole_0,d2_card_2,d5_card_1,t54_card_4,d4_wellord2,t7_card_4,t83_card_1,t14_card_1,t119_zfmisc_1,d5_partfun1,d4_tarski,d1_relat_1,d1_relat_1,d1_funct_1,d4_tarski,d4_tarski,d9_ordinal1,d9_xboole_0,d1_funct_1,d3_tarski,d4_funct_1,d4_tarski,d5_partfun1,t8_funct_1,d3_tarski,d5_funct_1,d4_funct_1,d4_tarski,d5_partfun1,t8_funct_1,d5_funct_1,d5_partfun1,d8_funct_1,t8_funct_1,d4_tarski,d4_tarski,d9_ordinal1,d9_xboole_0,t8_funct_1,d8_funct_1,d3_tarski,d4_funct_1,d4_tarski,t92_zfmisc_1,t25_relat_1,t8_funct_1,t119_zfmisc_1,d10_xboole_0,d3_tarski,t10_mcart_1,t23_mcart_1,d5_funct_1,d5_funct_1,t8_funct_1,d4_tarski,d4_tarski,d9_ordinal1,d9_xboole_0,t8_funct_1,d5_funct_1,t106_zfmisc_1,t92_zfmisc_1,t25_relat_1,t177_orders_1,d5_partfun1,d5_card_1,t2_card_4,d11_card_1,t11_card_4,t41_card_4,t72_card_4,t33_card_4,d5_card_1,d10_xboole_0,t7_xboole_1,t76_card_4,t27_card_1,t47_card_2,t79_xboole_1,d7_xboole_0,t113_zfmisc_1,t123_zfmisc_1,d7_xboole_0,t23_xboole_1,d7_xboole_0,t48_card_2,t48_card_2,t14_card_2,t14_card_2,t14_card_2,t54_card_4,d2_card_2,d2_card_2,d2_card_2,t67_card_4,t34_card_4,t21_card_1,d4_wellord2,t113_zfmisc_1,t123_zfmisc_1,t23_xboole_1,t123_zfmisc_1,t23_xboole_1,d7_xboole_0,t18_funct_4,t19_funct_4,t33_funct_4,d1_funct_4,t25_relat_1,t8_xboole_1,d10_xboole_0,t39_xboole_1,t120_zfmisc_1,t4_xboole_1,t120_zfmisc_1,t120_zfmisc_1,t39_xboole_1,t39_xboole_1,t8_xboole_1,t119_zfmisc_1,d5_partfun1,t79_xboole_1,d7_xboole_0,d8_funct_1,d2_xboole_0,d5_funct_1,d8_funct_1,t14_funct_4,t17_funct_4,d3_xboole_0,t26_funct_4,t4_card_2,d4_xboole_0,d2_xboole_0,t1_card_4,t27_card_1,t34_card_4,t13_card_1,t76_card_4,t26_card_1,d4_wellord2,t21_card_1,d2_card_2,t14_card_2,d4_wellord2,t33_card_4,t21_card_1,t79_xboole_1,t26_xboole_1,d7_xboole_0,t3_xboole_1,d7_xboole_0,t48_card_2,t14_card_2,d2_card_2,t40_xboole_1,t83_xboole_1,t7_xboole_1,d2_card_2,t7_xboole_1,t119_zfmisc_1,t126_zfmisc_1,t14_card_2,t1_xboole_1,t36_xboole_1,t27_card_1,d10_xboole_0,t21_card_1,d4_wellord2,t79_xboole_1,t33_funct_4,t19_funct_4,t25_relat_1,t18_funct_4,t8_xboole_1,d10_xboole_0,d1_funct_4,t39_xboole_1,t119_zfmisc_1,t12_xboole_1,t36_xboole_1,t1_xboole_1,t8_xboole_1,t119_zfmisc_1,d5_partfun1,t79_xboole_1,t26_xboole_1,d7_xboole_0,t3_xboole_1,d8_funct_1,d2_xboole_0,d5_funct_1,d8_funct_1,t14_funct_4,t17_funct_4,d3_xboole_0,t26_funct_4,t1_card_4,d3_tarski,d2_xboole_0,d4_xboole_0,d2_card_2,t14_card_2,d4_wellord2,t21_card_1]), [file(card_4,t77_card_4),interesting(1.00)]). fof(t87_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ( ~ v1_finset_1(A) => k1_card_1(A) = k1_card_1(k3_finseq_2(A)) ) ) ), inference(mizar_proof,[status(thm)],[s1_card_4,d1_card_4,t112_finseq_2,t1_card_4,t13_card_4,t86_card_4,t60_card_4,t128_finseq_2,t84_card_4,t92_zfmisc_1,t27_card_1,t56_card_4,d10_xboole_0]), [file(card_4,t87_card_4),interesting(0.96)]). fof(t82_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ! [D] : ( v1_card_1(D) => ( ( r2_hidden(A,B) & r2_hidden(C,D) ) => ( r2_hidden(k2_card_2(A,C),k2_card_2(B,D)) & r2_hidden(k2_card_2(C,A),k2_card_2(B,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t10_ordinal3,t26_card_4,d5_card_1,t73_card_1,t52_card_2,t66_xreal_1,t70_xreal_1,t2_xreal_1,t73_card_1,t11_card_4,t78_card_4,t83_card_1,t25_ordinal1,d9_xboole_0,t26_card_4,t79_card_4,d5_card_1,t52_card_2,d5_card_1,d3_tarski,t22_ordinal1,t25_ordinal1,d9_xboole_0]), [file(card_4,t82_card_4),interesting(0.83)]). fof(t32_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) => v4_ordinal1(A) ) ) ), inference(mizar_proof,[status(thm)],[t7_card_4,t83_card_1,t42_ordinal1,d8_xboole_0,t14_card_1,t83_card_1,t21_ordinal1,t34_ordinal1,t42_ordinal1,t19_ordinal2,t24_card_2,t24_card_2,t31_card_4,t10_ordinal1,t48_ordinal2,t21_card_1,d1_card_1,t7_ordinal1]), [file(card_4,t32_card_4),interesting(0.81)]). fof(t84_card_4,theorem,( ! [A] : ( ~ v1_finset_1(A) => k1_card_1(A) = k2_card_2(k3_card_1(0),k1_card_1(A)) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t11_card_4,t83_card_1,t78_card_4]), [file(card_4,t84_card_4),interesting(0.81)]). fof(t47_card_4,theorem,( ! [A,B] : ( ( v1_card_4(A) & v1_card_4(B) ) => v1_card_4(k2_xboole_0(A,B)) ) ), inference(mizar_proof,[status(thm)],[d1_card_4,t33_card_4,t41_card_4,t83_card_1,t47_card_2,t1_xboole_1,d1_card_4]), [file(card_4,t47_card_4),interesting(0.80)]). fof(t33_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) => k1_card_2(A,A) = A ) ) ), inference(mizar_proof,[status(thm)],[t32_card_4,t30_card_4,t25_card_2,t25_card_2,t53_ordinal2,t56_ordinal2,d1_card_2,d5_card_1]), [file(card_4,t33_card_4),interesting(0.79)]). fof(s1_card_4,theorem,( v1_card_4(a_0_0_card_4) ), inference(mizar_proof,[status(thm)],[s3_funct_1,d3_tarski,d5_funct_1,t45_card_4]), [file(card_4,s1_card_4),interesting(0.79)]). fof(s3_card_4,theorem,( v1_card_4(a_0_2_card_4) ), inference(mizar_proof,[status(thm)],[t53_card_4,t15_card_2,t53_card_4,t22_wellord2,d3_zfmisc_1,d4_wellord2,s3_funct_1,d3_tarski,d5_funct_1,t47_mcart_1,t50_mcart_1,d5_funct_1,t45_card_4]), [file(card_4,s3_card_4),interesting(0.78)]). fof(t79_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ~ ( ~ v1_finset_1(A) & ( r1_tarski(B,A) | r2_hidden(B,A) ) & ~ ( r1_tarski(k2_card_2(A,B),A) & r1_tarski(k2_card_2(B,A),A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t78_card_4,t19_card_4,t32_card_2,t10_ordinal3]), [file(card_4,t79_card_4),interesting(0.78)]). fof(s2_card_4,theorem,( v1_card_4(a_0_1_card_4) ), inference(mizar_proof,[status(thm)],[t53_card_4,d4_wellord2,s3_funct_1,d3_tarski,d5_funct_1,t7_mcart_1,d5_funct_1,t45_card_4]), [file(card_4,s2_card_4),interesting(0.77)]). fof(t43_card_4,theorem,( ! [A] : ( v1_finset_1(A) => v1_card_4(A) ) ), inference(mizar_proof,[status(thm)],[t74_card_1,d2_ordinal1,t21_card_1,t27_card_1,t83_card_1,t84_card_1,d1_card_4]), [file(card_4,t43_card_4),interesting(0.77)]). fof(t55_card_4,theorem,( ! [A,B] : ( ( v1_card_4(A) & v1_card_4(B) ) => v1_card_4(k2_zfmisc_1(A,B)) ) ), inference(mizar_proof,[status(thm)],[d1_card_4,t119_zfmisc_1,t27_card_1,d2_card_2,t54_card_4,t14_card_2,d1_card_4]), [file(card_4,t55_card_4),interesting(0.77)]). fof(t15_card_4,theorem,( ~ v1_finset_1(k5_numbers) ), inference(mizar_proof,[status(thm)],[t8_card_4]), [file(card_4,t15_card_4),interesting(0.75)]). fof(t46_card_4,theorem,( ! [A,B] : ( ( r1_tarski(A,B) & v1_card_4(B) ) => v1_card_4(A) ) ), inference(mizar_proof,[status(thm)],[t27_card_1,d1_card_4,t1_xboole_1,d1_card_4]), [file(card_4,t46_card_4),interesting(0.75)]). fof(t49_card_4,theorem,( ! [A,B] : ( v1_card_4(A) => v1_card_4(k4_xboole_0(A,B)) ) ), inference(mizar_proof,[status(thm)],[t36_xboole_1,t46_card_4]), [file(card_4,t49_card_4),interesting(0.75)]). fof(t80_card_4,theorem,( ! [A] : ( ~ v1_finset_1(A) => ( r2_wellord2(k2_zfmisc_1(A,A),A) & k1_card_1(k2_zfmisc_1(A,A)) = k1_card_1(A) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t77_card_4,d2_card_2,t14_card_2,t21_card_1]), [file(card_4,t80_card_4),interesting(0.75)]). fof(t1_card_4,theorem,( ! [A] : ( v1_finset_1(A) <=> v1_finset_1(k1_card_1(A)) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t68_card_1]), [file(card_4,t1_card_4),interesting(0.73)]). fof(t7_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => ( v1_finset_1(A) <=> r2_hidden(A,k5_ordinal2) ) ) ), inference(mizar_proof,[status(thm)],[t74_card_1,t6_card_4]), [file(card_4,t7_card_4),interesting(0.71)]). fof(t8_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => ( ~ v1_finset_1(A) <=> r1_ordinal1(k5_ordinal2,A) ) ) ), inference(mizar_proof,[status(thm)],[t7_card_4,t7_ordinal1,t26_ordinal1]), [file(card_4,t8_card_4),interesting(0.71)]). fof(t50_card_4,theorem,( ! [A,B] : ( ( v1_card_4(A) & v1_card_4(B) ) => v1_card_4(k5_xboole_0(A,B)) ) ), inference(mizar_proof,[status(thm)],[t49_card_4,t47_card_4]), [file(card_4,t50_card_4),interesting(0.71)]). fof(t65_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ( v1_card_4(A) => v1_card_4(k3_finseq_2(A)) ) ) ), inference(mizar_proof,[status(thm)],[s2_funct_1,t58_card_4,t128_finseq_2,d4_card_3,d3_tarski,d4_tarski,d5_funct_1,d4_tarski,d10_xboole_0,d3_tarski,d4_tarski,d5_funct_1,d4_tarski,t44_card_4,t61_card_4]), [file(card_4,t65_card_4),interesting(0.70)]). fof(t19_card_4,theorem,( ! [A] : ( v1_card_1(A) => r1_tarski(0,A) ) ), inference(mizar_proof,[status(thm)],[t10_ordinal3,t13_card_1]), [file(card_4,t19_card_4),interesting(0.70)]). fof(t44_card_4,theorem, ( v1_card_4(k5_ordinal2) & v1_card_4(k5_numbers) ), inference(mizar_proof,[status(thm)],[t83_card_1,t84_card_1,d1_card_4,t83_card_1,t84_card_1,d1_card_4]), [file(card_4,t44_card_4),interesting(0.69)]). fof(t20_card_4,theorem,( ! [A,B] : ( k1_card_1(A) = k1_card_1(B) <=> k2_card_1(A) = k2_card_1(B) ) ), inference(mizar_proof,[status(thm)],[t34_card_1,t24_ordinal1,d6_card_1,t22_ordinal1]), [file(card_4,t20_card_4),interesting(0.68)]). fof(t85_card_4,theorem,( ! [A,B] : ( v1_finset_1(A) => ( A = k1_xboole_0 | v1_finset_1(B) | k2_card_2(k1_card_1(B),k1_card_1(A)) = k1_card_1(B) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t59_card_2,t13_card_4,t10_ordinal3,t78_card_4]), [file(card_4,t85_card_4),interesting(0.68)]). fof(t13_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( v1_finset_1(A) => ( v1_finset_1(B) | ( r2_hidden(A,B) & r1_tarski(A,B) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t7_card_4,t11_card_4,t83_card_1,t13_card_1]), [file(card_4,t13_card_4),interesting(0.66)]). fof(t72_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r1_tarski(A,k1_card_2(A,B)) & r1_tarski(B,k1_card_2(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t27_ordinal3,t27_card_1,d1_card_2]), [file(card_4,t72_card_4),interesting(0.66)]). fof(t74_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ! [D] : ( v1_card_1(D) => ( ( r2_hidden(A,B) & r2_hidden(C,D) ) => ( r2_hidden(k1_card_2(A,C),k1_card_2(B,D)) & r2_hidden(k1_card_2(C,A),k1_card_2(B,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t26_card_4,d5_card_1,t73_card_1,t51_card_2,t10_xreal_1,t73_card_1,t11_card_4,t34_card_4,t83_card_1,t25_ordinal1,d9_xboole_0,t26_card_4,t34_card_4,d5_card_1,t51_card_2,d5_card_1,d3_tarski,t34_card_4,t25_ordinal1,d9_xboole_0]), [file(card_4,t74_card_4),interesting(0.65)]). fof(t22_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(A,B) <=> r1_tarski(k2_card_1(A),B) ) ) ) ), inference(mizar_proof,[status(thm)],[t32_card_1,d5_card_1,d6_card_1]), [file(card_4,t22_card_4),interesting(0.64)]). fof(t11_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) <=> r1_ordinal1(k3_card_1(0),A) ) ) ), inference(mizar_proof,[status(thm)],[t7_card_4,t14_card_1,t83_card_1]), [file(card_4,t11_card_4),interesting(0.64)]). fof(t78_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(0,B) => ( v1_finset_1(A) | ( ~ r1_tarski(B,A) & ~ r2_hidden(B,A) ) | ( k2_card_2(A,B) = A & k2_card_2(B,A) = A ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t77_card_4,t24_card_4,t13_card_1,t68_card_4,t33_card_2,d10_xboole_0]), [file(card_4,t78_card_4),interesting(0.64)]). fof(t62_card_4,theorem,( ! [A] : ( ( v1_card_4(A) & ! [B] : ( r2_hidden(B,A) => v1_card_4(B) ) ) => v1_card_4(k3_tarski(A)) ) ), inference(mizar_proof,[status(thm)],[d1_card_4,d1_card_4,t54_card_4,t60_card_4,d1_card_4]), [file(card_4,t62_card_4),interesting(0.63)]). fof(t54_card_4,theorem,( k2_card_2(k3_card_1(0),k3_card_1(0)) = k3_card_1(0) ), inference(mizar_proof,[status(thm)],[t53_card_4,t83_card_1,t84_card_1,d2_card_2]), [file(card_4,t54_card_4),interesting(0.62)]). fof(t5_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => k4_xboole_0(k1_ordinal1(A),k1_tarski(A)) = A ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,t13_ordinal1,d1_tarski,d10_xboole_0,d3_tarski,t13_ordinal1,d1_tarski,d4_xboole_0]), [file(card_4,t5_card_4),interesting(0.61)]). fof(t26_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( v1_finset_1(A) => ( ( ~ r1_tarski(B,A) & ~ r2_hidden(B,A) ) | v1_finset_1(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_card_1,t13_finset_1]), [file(card_4,t26_card_4),interesting(0.61)]). fof(t39_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( ~ v1_finset_1(A) => ( ~ v1_finset_1(k1_card_2(A,B)) & ~ v1_finset_1(k1_card_2(B,A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_ordinal1,d9_xboole_0,t26_card_4,t34_card_4,t34_card_4]), [file(card_4,t39_card_4),interesting(0.61)]). fof(t48_card_4,theorem,( ! [A,B] : ( v1_card_4(A) => ( v1_card_4(k3_xboole_0(A,B)) & v1_card_4(k3_xboole_0(B,A)) ) ) ), inference(mizar_proof,[status(thm)],[t17_xboole_1,t46_card_4]), [file(card_4,t48_card_4),interesting(0.61)]). fof(t58_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( v1_card_4(A) => v1_card_4(k4_finseq_2(B,A)) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_card_4,t56_card_4,d1_card_4,t43_card_4,t112_finseq_2,t55_card_4,t57_card_4,d1_card_4,t21_card_1,d1_card_4,s1_nat_1]), [file(card_4,t58_card_4),interesting(0.61)]). fof(t81_card_4,theorem,( ! [A,B] : ( v1_finset_1(B) => ( v1_finset_1(A) | B = k1_xboole_0 | ( r2_wellord2(k2_zfmisc_1(A,B),A) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(A) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t59_card_2,t13_card_4,t10_ordinal3,t78_card_4,d2_card_2,t14_card_2,t21_card_1]), [file(card_4,t81_card_4),interesting(0.61)]). fof(t31_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => ( r1_ordinal1(k5_ordinal2,A) => k14_ordinal2(k4_ordinal2,A) = A ) ) ), inference(mizar_proof,[status(thm)],[s3_ordinal2,t19_ordinal2,t46_ordinal2,t35_ordinal2,d1_zf_refle,t29_ordinal2,d5_funct_1,t23_ordinal1,t65_card_1,t7_card_4,d5_card_1,d4_ordinal2,t49_card_2,t41_ordinal1,t19_ordinal2,t48_ordinal2,t22_ordinal1,t30_ordinal3,t27_ordinal3,d10_xboole_0,t33_ordinal3]), [file(card_4,t31_card_4),interesting(0.60)]). fof(t30_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => ( v4_ordinal1(A) => k15_ordinal2(A,k1_ordinal1(k4_ordinal2)) = A ) ) ), inference(mizar_proof,[status(thm)],[t29_card_4,t48_ordinal3,t22_nat_1,t52_card_1,t45_ordinal2,t42_ordinal1,t44_ordinal2]), [file(card_4,t30_card_4),interesting(0.60)]). fof(t6_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v3_ordinal1(B) => ( r2_wellord2(B,A) => B = A ) ) ) ), inference(mizar_proof,[status(thm)],[t46_card_1,t52_card_1,t46_card_1,t10_ordinal3,d5_card_1,d5_ordinal2,t21_card_1,t27_card_1,t84_card_1,t14_card_1,t42_ordinal1,t10_ordinal1,t5_card_4,t61_card_1,s1_nat_1]), [file(card_4,t6_card_4),interesting(0.59)]). fof(t76_card_4,theorem,( ! [A,B] : ( ( k1_card_2(k1_card_1(A),k1_card_1(B)) = k1_card_1(A) & r2_hidden(k1_card_1(B),k1_card_1(A)) ) => k1_card_1(k4_xboole_0(A,B)) = k1_card_1(A) ) ), inference(mizar_proof,[status(thm)],[t7_xboole_1,t27_card_1,t47_card_2,d10_xboole_0,t39_xboole_1,t79_xboole_1,t48_card_2,t72_card_4,t33_card_4,t13_card_1,t74_card_4,t26_card_4,t1_card_4,d11_card_1,t51_card_2,t71_card_1,t59_card_2]), [file(card_4,t76_card_4),interesting(0.59)]). fof(t2_card_4,theorem,( ! [A] : ( v1_finset_1(A) <=> r2_hidden(k1_card_1(A),k3_card_1(0)) ) ), inference(mizar_proof,[status(thm)],[t83_card_1,t84_card_1,t58_card_3,t83_card_1,t1_card_4]), [file(card_4,t2_card_4),interesting(0.58)]). fof(t24_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( r2_hidden(0,A) <=> r1_tarski(1,A) ) ) ), inference(mizar_proof,[status(thm)],[t76_card_1,l1_card_4,t22_card_4]), [file(card_4,t24_card_4),interesting(0.58)]). fof(t21_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( A = B <=> k2_card_1(B) = k2_card_1(A) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t20_card_4]), [file(card_4,t21_card_4),interesting(0.56)]). fof(t34_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ~ ( ~ v1_finset_1(A) & ( r1_tarski(B,A) | r2_hidden(B,A) ) & ~ ( k1_card_2(A,B) = A & k1_card_2(B,A) = A ) ) ) ) ), inference(mizar_proof,[status(thm)],[t33_card_4,t13_card_1,d1_card_2,d5_card_1,t50_ordinal2,t27_ordinal3,t27_card_1,d10_xboole_0]), [file(card_4,t34_card_4),interesting(0.56)]). fof(t38_card_4,theorem,( ! [A] : ( ( v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( v1_finset_1(B) & v1_card_1(B) ) => v1_finset_1(k1_card_2(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t51_card_2]), [file(card_4,t38_card_4),interesting(0.54)]). fof(t40_card_4,theorem,( ! [A] : ( ( v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( v1_finset_1(B) & v1_card_1(B) ) => v1_finset_1(k2_card_2(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t52_card_2]), [file(card_4,t40_card_4),interesting(0.54)]). fof(t75_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ( r2_hidden(k1_card_2(A,B),k1_card_2(A,C)) => r2_hidden(B,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_card_1,t41_card_4,t14_card_1]), [file(card_4,t75_card_4),interesting(0.54)]). fof(t83_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ( r2_hidden(k2_card_2(A,B),k2_card_2(A,C)) => r2_hidden(B,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_card_1,t68_card_4,t14_card_1]), [file(card_4,t83_card_4),interesting(0.54)]). fof(t23_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(A,k2_card_1(B)) <=> r1_tarski(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_card_1,t22_card_4]), [file(card_4,t23_card_4),interesting(0.52)]). fof(t9_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( v1_finset_1(A) <=> r2_hidden(A,k3_card_1(0)) ) ) ), inference(mizar_proof,[status(thm)],[t7_card_4,t83_card_1]), [file(card_4,t9_card_4),interesting(0.52)]). fof(t17_card_4,theorem,( ! [A] : ( A = k1_xboole_0 <=> k1_card_1(A) = 0 ) ), inference(mizar_proof,[status(thm)],[t47_card_1,t59_card_2]), [file(card_4,t17_card_4),interesting(0.51)]). fof(t36_card_4,theorem,( ! [A,B] : ( v1_finset_1(B) => ( v1_finset_1(A) | ( r2_wellord2(k2_xboole_0(A,B),A) & k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t13_card_4,t7_xboole_1,t34_card_4,t27_card_1,t47_card_2,d10_xboole_0,t21_card_1]), [file(card_4,t36_card_4),interesting(0.51)]). fof(t60_card_4,theorem,( ! [A,B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ( ( r1_tarski(k1_card_1(A),B) & ! [D] : ( r2_hidden(D,A) => r1_tarski(k1_card_1(D),C) ) ) => r1_tarski(k1_card_1(k3_tarski(A)),k2_card_2(B,C)) ) ) ) ), inference(mizar_proof,[status(thm)],[t71_relat_1,t35_funct_1,t59_card_4,d4_card_3,t71_relat_1]), [file(card_4,t60_card_4),interesting(0.49)]). fof(t35_card_4,theorem,( ! [A,B] : ~ ( ~ v1_finset_1(A) & ( r2_wellord2(A,B) | r2_wellord2(B,A) ) & ~ ( r2_wellord2(k2_xboole_0(A,B),A) & k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t21_card_1,t7_xboole_1,t33_card_4,t27_card_1,t47_card_2,d10_xboole_0,t21_card_1]), [file(card_4,t35_card_4),interesting(0.48)]). fof(t56_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ( r2_wellord2(k4_finseq_2(1,A),A) & k1_card_1(k4_finseq_2(1,A)) = k1_card_1(A) ) ) ), inference(mizar_proof,[status(thm)],[s3_funct_1,d4_wellord2,d8_funct_1,d8_finseq_1,d3_tarski,d5_funct_1,t116_finseq_2,d10_xboole_0,d3_tarski,t117_finseq_2,d5_funct_1,t21_card_1]), [file(card_4,t56_card_4),interesting(0.48)]). fof(t61_card_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( ( v1_card_4(k1_relat_1(A)) & ! [B] : ( r2_hidden(B,k1_relat_1(A)) => v1_card_4(k1_funct_1(A,B)) ) ) => v1_card_4(k3_card_3(A)) ) ) ), inference(mizar_proof,[status(thm)],[d1_card_4,d1_card_4,t54_card_4,t59_card_4,d1_card_4]), [file(card_4,t61_card_4),interesting(0.47)]). fof(l1_card_4,theorem, ( 0 = k1_card_1(0) & 1 = k1_card_1(1) & 2 = k1_card_1(2) ), inference(mizar_proof,[status(thm)],[d5_card_1]), [file(card_4,l1_card_4),interesting(0.45)]). fof(t25_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( r2_hidden(1,A) <=> r1_tarski(2,A) ) ) ), inference(mizar_proof,[status(thm)],[t76_card_1,l1_card_4,t22_card_4]), [file(card_4,t25_card_4),interesting(0.45)]). fof(t27_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => ( v4_ordinal1(A) <=> ! [B] : ( v3_ordinal1(B) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(B,A) => r2_hidden(k14_ordinal2(B,C),A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t44_ordinal2,t52_card_1,t45_ordinal2,t41_ordinal1,s1_nat_1,t48_ordinal2,d4_ordinal2,t41_ordinal1]), [file(card_4,t27_card_4),interesting(0.43)]). fof(t86_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( ~ v1_finset_1(A) & B != 0 & ~ ( r2_wellord2(k4_finseq_2(B,A),A) & k1_card_1(k4_finseq_2(B,A)) = k1_card_1(A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t57_card_4,t14_card_2,t56_card_4,d2_card_2,t112_finseq_2,t59_card_2,t1_card_4,t13_card_4,t10_ordinal3,t78_card_4,s1_nat_1,t21_card_1]), [file(card_4,t86_card_4),interesting(0.43)]). fof(t73_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( A != 0 => ( r1_tarski(B,k2_card_2(B,A)) & r1_tarski(B,k2_card_2(A,B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t25_card_2,t39_ordinal3,t27_card_1]), [file(card_4,t73_card_4),interesting(0.40)]). fof(t66_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => r1_tarski(k3_card_1(0),k1_card_1(k3_finseq_2(A))) ) ), inference(mizar_proof,[status(thm)],[s2_funct_1,t66_finseq_1,t25_finseq_1,d12_funct_1,d12_funct_1,d11_finseq_1,d11_finseq_1,t35_finseq_1,t57_finseq_1,d12_funct_1,s1_nat_1,d3_tarski,t83_card_1,t84_card_1,t2_card_2]), [file(card_4,t66_card_4),interesting(0.39)]). fof(t4_card_4,theorem,( ! [A] : ( v1_finset_1(A) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & k1_card_1(A) = k1_card_1(B) ) ) ), inference(mizar_proof,[status(thm)],[d11_card_1,t1_card_4]), [file(card_4,t4_card_4),interesting(0.36)]). fof(t45_card_4,theorem,( ! [A] : ( v1_card_4(A) <=> ? [B] : ( v1_relat_1(B) & v1_funct_1(B) & k1_relat_1(B) = k5_numbers & r1_tarski(A,k2_relat_1(B)) ) ) ), inference(mizar_proof,[status(thm)],[d1_card_4,t28_card_1,t83_card_1,t84_card_1,t28_card_1,t83_card_1,t84_card_1,d1_card_4]), [file(card_4,t45_card_4),interesting(0.35)]). fof(t29_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & k15_ordinal2(A,k1_ordinal1(k4_ordinal2)) = k14_ordinal2(A,B) ) ) ), inference(mizar_proof,[status(thm)],[t44_ordinal2,t52_ordinal2,t51_card_1,t53_ordinal2,t45_ordinal2,t48_ordinal2,t45_ordinal2,t52_card_1,t45_ordinal2,t28_card_4,s3_ordinal2,t54_ordinal2,t35_ordinal2,d1_zf_refle,d4_tarski,t23_ordinal1,t29_ordinal2,d5_funct_1,t23_ordinal1,t27_card_4,t22_ordinal1,t44_ordinal2,t19_ordinal1,d10_xboole_0,t10_ordinal1,d2_ordinal1,t44_ordinal2,t56_ordinal2,t59_ordinal2,s1_ordinal2]), [file(card_4,t29_card_4),interesting(0.34)]). fof(t3_card_4,theorem,( ! [A] : ( v1_finset_1(A) => ( r2_hidden(k1_card_1(A),k3_card_1(0)) & r2_hidden(k1_card_1(A),k5_ordinal2) ) ) ), inference(mizar_proof,[status(thm)],[t2_card_4,t83_card_1]), [file(card_4,t3_card_4),interesting(0.32)]). fof(t67_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_tarski(k2_card_2(k3_card_1(0),k1_card_1(A)),k3_card_1(0)) & r1_tarski(k2_card_2(k1_card_1(A),k3_card_1(0)),k3_card_1(0)) ) ) ), inference(mizar_proof,[status(thm)],[l1_card_4,t32_card_2,t2_xboole_1,t11_card_4,d5_card_1,t52_card_1,d5_card_1,t83_card_1,t84_card_1,t25_card_2,t53_ordinal2,t83_card_1,t84_card_1,t24_card_2,t83_card_1,t84_card_1,t25_card_2,t34_card_4,s1_nat_1]), [file(card_4,t67_card_4),interesting(0.32)]). fof(t63_card_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( ( v1_finset_1(k1_relat_1(A)) & ! [B] : ( r2_hidden(B,k1_relat_1(A)) => v1_finset_1(k1_funct_1(A,B)) ) ) => v1_finset_1(k3_card_3(A)) ) ) ), inference(mizar_proof,[status(thm)],[d11_card_1,t1_xboole_1,s3_card_3,t59_card_4,d11_card_1,t52_card_2,t26_card_4,t1_card_4,t14_card_3,t64_relat_1]), [file(card_4,t63_card_4),interesting(0.31)]). fof(t14_card_4,theorem,( ! [A] : ( ~ ( ~ v1_finset_1(A) & ! [B] : ~ ( r1_tarski(B,A) & k1_card_1(B) = k3_card_1(0) ) ) & ~ ( ? [B] : ( r1_tarski(B,A) & k1_card_1(B) = k3_card_1(0) ) & v1_finset_1(A) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t7_card_4,t83_card_1,t14_card_1,d5_card_1,d4_wellord2,t144_relat_1,d4_wellord2,t84_funct_1,t91_relat_1,t148_relat_1,d5_card_1,t27_card_1,t14_card_1,t2_card_4]), [file(card_4,t14_card_4),interesting(0.30)]). fof(t71_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ~ ( ( r2_hidden(A,B) | r1_tarski(A,B) ) & C != 0 & ~ ( r1_tarski(k3_card_2(C,A),k3_card_2(C,B)) & r1_tarski(k3_card_2(A,C),k3_card_2(B,C)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t70_card_4,t39_card_2,t19_card_4,t70_card_4]), [file(card_4,t71_card_4),interesting(0.30)]). fof(t37_card_4,theorem,( ! [A,B] : ~ ( ~ v1_finset_1(A) & ( r2_hidden(k1_card_1(B),k1_card_1(A)) | r1_tarski(k1_card_1(B),k1_card_1(A)) ) & ~ ( r2_wellord2(k2_xboole_0(A,B),A) & k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ), inference(mizar_proof,[status(thm)],[t1_card_4,t13_card_1,t7_xboole_1,t34_card_4,t27_card_1,t47_card_2,d10_xboole_0,t21_card_1]), [file(card_4,t37_card_4),interesting(0.27)]). fof(t28_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v3_ordinal1(B) => ( k14_ordinal2(B,k1_ordinal1(A)) = k14_ordinal2(k1_ordinal1(B),A) & k14_ordinal2(B,k1_nat_1(A,1)) = k14_ordinal2(k1_ordinal1(B),A) ) ) ) ), inference(mizar_proof,[status(thm)],[t48_ordinal2,d4_ordinal2,t44_ordinal2,t52_card_1,t45_ordinal2,t48_ordinal2,t33_ordinal3,t48_ordinal2,s1_nat_1,t52_card_1]), [file(card_4,t28_card_4),interesting(0.22)]). fof(t59_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( r1_tarski(k1_card_1(k1_relat_1(C)),A) & ! [D] : ( r2_hidden(D,k1_relat_1(C)) => r1_tarski(k1_card_1(k1_funct_1(C,D)),B) ) ) => r1_tarski(k1_card_1(k3_card_3(C)),k2_card_2(A,B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t54_card_3,d2_card_3,t19_funcop_1,d2_card_3,t13_funcop_1,t43_card_3,t1_xboole_1,t118_zfmisc_1,d7_card_3,t36_card_3,t14_card_2,t11_card_2,t27_card_1,d2_card_2,t1_xboole_1]), [file(card_4,t59_card_4),interesting(0.14)]). fof(t57_card_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_wellord2(k2_zfmisc_1(k4_finseq_2(B,A),k4_finseq_2(C,A)),k4_finseq_2(k1_nat_1(B,C),A)) & k1_card_1(k2_zfmisc_1(k4_finseq_2(B,A),k4_finseq_2(C,A))) = k1_card_1(k4_finseq_2(k1_nat_1(B,C),A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t33_zfmisc_1,t10_mcart_1,t23_mcart_1,t10_mcart_1,t10_mcart_1,s2_funct_1,d4_wellord2,d8_funct_1,t109_finseq_2,t64_finseq_1,t64_finseq_1,t35_finseq_1,t35_finseq_1,t23_nat_1,t25_finseq_1,t47_finseq_1,t46_finseq_1,d3_tarski,d5_funct_1,t127_finseq_2,d10_xboole_0,d3_tarski,t126_finseq_2,t33_zfmisc_1,d5_funct_1,t21_card_1]), [file(card_4,t57_card_4),interesting(0.09)]). fof(l54_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( k2_nat_1(k1_card_4(2,A),k1_nat_1(k2_nat_1(2,B),1)) = k2_nat_1(k1_card_4(2,C),k1_nat_1(k2_nat_1(2,D),1)) => r1_xreal_0(A,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t28_nat_1,t22_nat_1,t13_newton,t51_card_4,t5_xcmplx_1,t13_newton,t10_newton,d3_nat_1,t57_nat_1,t54_nat_1]), [file(card_4,l54_card_4),interesting(0.05)]). fof(s8_funct_2,theorem,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k2_zfmisc_1(f1_s8_funct_2,f2_s8_funct_2),f3_s8_funct_2) & m2_relset_1(A,k2_zfmisc_1(f1_s8_funct_2,f2_s8_funct_2),f3_s8_funct_2) & ! [B] : ( m1_subset_1(B,f1_s8_funct_2) => ! [C] : ( m1_subset_1(C,f2_s8_funct_2) => k1_funct_1(A,k4_tarski(B,C)) = f4_s8_funct_2(B,C) ) ) ) ), file(funct_2,s8_funct_2), [interesting(0.00)]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [interesting(0.00)]). fof(t12_relset_1,theorem,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) ) ), file(relset_1,t12_relset_1), [interesting(0.00)]). fof(d8_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), [interesting(0.00)]). fof(t10_mcart_1,theorem,( ! [A,B,C] : ( r2_hidden(A,k2_zfmisc_1(B,C)) => ( r2_hidden(k1_mcart_1(A),B) & r2_hidden(k2_mcart_1(A),C) ) ) ), file(mcart_1,t10_mcart_1), [interesting(0.00)]). fof(t23_mcart_1,theorem,( ! [A,B,C] : ( r2_hidden(A,k2_zfmisc_1(B,C)) => A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ), file(mcart_1,t23_mcart_1), [interesting(0.00)]). fof(l56_card_4,theorem,( ! [A] : ~ ( r2_hidden(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => A != k1_domain_1(k5_numbers,k5_numbers,B,C) ) ) ) ), inference(mizar_proof,[status(thm)],[t10_mcart_1,t23_mcart_1]), [file(card_4,l56_card_4),interesting(0.00)]). fof(t28_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => B != k2_xcmplx_0(A,C) ) ) ) ) ), file(nat_1,t28_nat_1), [interesting(0.00)]). fof(t22_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ~ ( A != 0 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => A != k1_nat_1(B,1) ) ) ) ), file(nat_1,t22_nat_1), [interesting(0.00)]). fof(t13_newton,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v1_xreal_0(C) => k2_newton(C,k2_xcmplx_0(A,B)) = k3_xcmplx_0(k2_newton(C,A),k2_newton(C,B)) ) ) ) ), file(newton,t13_newton), [interesting(0.00)]). fof(t9_newton,theorem,( ! [A] : ( v1_xreal_0(A) => k2_newton(A,0) = 1 ) ), file(newton,t9_newton), [interesting(0.00)]). fof(t11_newton,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_xreal_0(B) => k2_newton(B,k2_xcmplx_0(A,1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) ), file(newton,t11_newton), [interesting(0.00)]). fof(t6_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ~ ( k3_xcmplx_0(A,B) = 0 & A != 0 & B != 0 ) ) ) ), file(xcmplx_1,t6_xcmplx_1), [interesting(0.00)]). fof(s1_nat_1,theorem, ( ( p1_s1_nat_1(0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( p1_s1_nat_1(A) => p1_s1_nat_1(k1_nat_1(A,1)) ) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => p1_s1_nat_1(A) ) ), file(nat_1,s1_nat_1), [interesting(0.00)]). fof(t51_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ( ~ ( ~ ( B = 0 & A != 0 ) & k3_newton(B,A) = 0 ) & ~ ( k3_newton(B,A) != 0 & B = 0 & A != 0 ) ) ) ) ), inference(mizar_proof,[status(thm)],[t9_newton,t11_newton,t6_xcmplx_1,s1_nat_1]), [file(card_4,t51_card_4),interesting(0.00)]). fof(t5_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => ( k3_xcmplx_0(B,A) = k3_xcmplx_0(C,A) => ( A = 0 | B = C ) ) ) ) ) ), file(xcmplx_1,t5_xcmplx_1), [interesting(0.00)]). fof(t10_newton,theorem,( ! [A] : ( v1_xreal_0(A) => k2_newton(A,1) = A ) ), file(newton,t10_newton), [interesting(0.00)]). fof(d3_nat_1,definition,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_nat_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & B = k3_xcmplx_0(A,C) ) ) ) ) ), file(nat_1,d3_nat_1), [interesting(0.00)]). fof(t57_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( ( r1_nat_1(A,B) & r1_nat_1(A,k2_xcmplx_0(B,C)) ) => r1_nat_1(A,C) ) ) ) ) ), file(nat_1,t57_nat_1), [interesting(0.00)]). fof(t54_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_nat_1(B,A) => ( r1_xreal_0(A,0) | r1_xreal_0(B,A) ) ) ) ) ), file(nat_1,t54_nat_1), [interesting(0.00)]). fof(t1_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,A) ) => A = B ) ) ) ), file(xreal_1,t1_xreal_1), [interesting(0.00)]). fof(t52_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( k2_nat_1(k1_card_4(2,A),k1_nat_1(k2_nat_1(2,B),1)) = k2_nat_1(k1_card_4(2,C),k1_nat_1(k2_nat_1(2,D),1)) => ( A = C & B = D ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[l54_card_4,t1_xreal_1,t51_card_4,t5_xcmplx_1]), [file(card_4,t52_card_4),interesting(0.00)]). fof(t26_card_1,theorem,( ! [A,B] : ( r1_tarski(k1_card_1(A),k1_card_1(B)) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & v2_funct_1(C) & k1_relat_1(C) = A & r1_tarski(k2_relat_1(C),B) ) ) ), file(card_1,t26_card_1), [interesting(0.00)]). fof(t118_zfmisc_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => ( r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C)) & r1_tarski(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ) ), file(zfmisc_1,t118_zfmisc_1), [interesting(0.00)]). fof(t27_card_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => r1_tarski(k1_card_1(A),k1_card_1(B)) ) ), file(card_1,t27_card_1), [interesting(0.00)]). fof(t13_card_2,theorem,( ! [A,B] : ( r2_wellord2(A,k2_zfmisc_1(A,k1_tarski(B))) & k1_card_1(A) = k1_card_1(k2_zfmisc_1(A,k1_tarski(B))) ) ), file(card_2,t13_card_2), [interesting(0.00)]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.00)]). fof(t21_card_1,theorem,( ! [A,B] : ( r2_wellord2(A,B) <=> k1_card_1(A) = k1_card_1(B) ) ), file(card_1,t21_card_1), [interesting(0.00)]). fof(t53_card_4,theorem, ( r2_wellord2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k5_numbers) & k1_card_1(k5_numbers) = k1_card_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ), inference(mizar_proof,[status(thm)],[s8_funct_2,d1_funct_2,t12_relset_1,d8_funct_1,l56_card_4,l56_card_4,t52_card_4,t26_card_1,t118_zfmisc_1,t27_card_1,t13_card_2,d10_xboole_0,t21_card_1]), [file(card_4,t53_card_4),interesting(0.00)]). fof(d4_wellord2,definition,( ! [A,B] : ( r2_wellord2(A,B) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & v2_funct_1(C) & k1_relat_1(C) = A & k2_relat_1(C) = B ) ) ), file(wellord2,d4_wellord2), [interesting(0.00)]). fof(s3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s3_funct_1 & ! [B] : ( r2_hidden(B,f1_s3_funct_1) => k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) ), file(funct_1,s3_funct_1), [interesting(0.00)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.00)]). fof(d5_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), [interesting(0.00)]). fof(t7_mcart_1,theorem,( ! [A,B] : ( k1_mcart_1(k4_tarski(A,B)) = A & k2_mcart_1(k4_tarski(A,B)) = B ) ), file(mcart_1,t7_mcart_1), [interesting(0.00)]). fof(d1_card_4,definition,( ! [A] : ( v1_card_4(A) <=> r1_tarski(k1_card_1(A),k3_card_1(0)) ) ), file(card_4,d1_card_4), [interesting(0.00)]). fof(t28_card_1,theorem,( ! [A,B] : ( r1_tarski(k1_card_1(A),k1_card_1(B)) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & k1_relat_1(C) = B & r1_tarski(A,k2_relat_1(C)) ) ) ), file(card_1,t28_card_1), [interesting(0.00)]). fof(t83_card_1,theorem,( k3_card_1(0) = k5_ordinal2 ), file(card_1,t83_card_1), [interesting(0.00)]). fof(t84_card_1,theorem,( k1_card_1(k5_ordinal2) = k5_ordinal2 ), file(card_1,t84_card_1), [interesting(0.00)]). fof(t15_card_2,theorem,( ! [A,B,C,D] : ( ( r2_wellord2(A,B) & r2_wellord2(C,D) ) => ( r2_wellord2(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) & k1_card_1(k2_zfmisc_1(A,C)) = k1_card_1(k2_zfmisc_1(B,D)) ) ) ), file(card_2,t15_card_2), [interesting(0.00)]). fof(t22_wellord2,theorem,( ! [A,B,C] : ( ( r2_wellord2(A,B) & r2_wellord2(B,C) ) => r2_wellord2(A,C) ) ), file(wellord2,t22_wellord2), [interesting(0.00)]). fof(d3_zfmisc_1,definition,( ! [A,B,C] : k3_zfmisc_1(A,B,C) = k2_zfmisc_1(k2_zfmisc_1(A,B),C) ), file(zfmisc_1,d3_zfmisc_1), [interesting(0.00)]). fof(t47_mcart_1,theorem,( ! [A,B,C] : ~ ( A != k1_xboole_0 & B != k1_xboole_0 & C != k1_xboole_0 & ? [D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) & ? [E,F,G] : ( D = k3_mcart_1(E,F,G) & ~ ( k5_mcart_1(A,B,C,D) = E & k6_mcart_1(A,B,C,D) = F & k7_mcart_1(A,B,C,D) = G ) ) ) ) ), file(mcart_1,t47_mcart_1), [interesting(0.00)]). fof(t50_mcart_1,theorem,( ! [A,B,C] : ~ ( A != k1_xboole_0 & B != k1_xboole_0 & C != k1_xboole_0 & ~ ! [D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => ( k5_mcart_1(A,B,C,D) = k1_mcart_1(k1_mcart_1(D)) & k6_mcart_1(A,B,C,D) = k2_mcart_1(k1_mcart_1(D)) & k7_mcart_1(A,B,C,D) = k2_mcart_1(D) ) ) ) ), file(mcart_1,t50_mcart_1), [interesting(0.00)]). fof(t10_card_4,theorem,( $true ), file(card_4,t10_card_4), [interesting(0.00)]). fof(t12_card_4,theorem,( $true ), file(card_4,t12_card_4), [interesting(0.00)]). fof(d5_card_1,definition,( ! [A,B] : ( v1_card_1(B) => ( B = k1_card_1(A) <=> r2_wellord2(A,B) ) ) ), file(card_1,d5_card_1), [interesting(0.00)]). fof(t68_card_1,theorem,( ! [A,B] : ( ( r2_wellord2(A,B) & v1_finset_1(A) ) => v1_finset_1(B) ) ), file(card_1,t68_card_1), [interesting(0.00)]). fof(t74_card_1,theorem,( ! [A] : ~ ( v1_finset_1(A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ r2_wellord2(A,B) ) ) ), file(card_1,t74_card_1), [interesting(0.00)]). fof(t46_card_1,theorem,( ! [A] : ( r2_wellord2(A,k1_xboole_0) <=> A = k1_xboole_0 ) ), file(card_1,t46_card_1), [interesting(0.00)]). fof(t52_card_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_ordinal1(A) = k1_nat_1(A,1) ) ), file(card_1,t52_card_1), [interesting(0.00)]). fof(t10_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ( A != k1_xboole_0 => r2_hidden(k1_xboole_0,A) ) ) ), file(ordinal3,t10_ordinal3), [interesting(0.00)]). fof(d5_ordinal2,definition,( ! [A] : ( A = k5_ordinal2 <=> ( r2_hidden(k1_xboole_0,A) & v4_ordinal1(A) & v3_ordinal1(A) & ! [B] : ( v3_ordinal1(B) => ( ( r2_hidden(k1_xboole_0,B) & v4_ordinal1(B) ) => r1_tarski(A,B) ) ) ) ) ), file(ordinal2,d5_ordinal2), [interesting(0.00)]). fof(t14_card_1,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(A,B) <=> ~ r1_tarski(B,A) ) ) ) ), file(card_1,t14_card_1), [interesting(0.00)]). fof(t42_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ( ~ ( ~ v4_ordinal1(A) & ! [B] : ( v3_ordinal1(B) => A != k1_ordinal1(B) ) ) & ~ ( ? [B] : ( v3_ordinal1(B) & A = k1_ordinal1(B) ) & v4_ordinal1(A) ) ) ) ), file(ordinal1,t42_ordinal1), [interesting(0.00)]). fof(t10_ordinal1,theorem,( ! [A] : r2_hidden(A,k1_ordinal1(A)) ), file(ordinal1,t10_ordinal1), [interesting(0.00)]). fof(d4_xboole_0,definition,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), [interesting(0.00)]). fof(t13_ordinal1,theorem,( ! [A,B] : ( r2_hidden(A,k1_ordinal1(B)) <=> ( r2_hidden(A,B) | A = B ) ) ), file(ordinal1,t13_ordinal1), [interesting(0.00)]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.00)]). fof(t61_card_1,theorem,( ! [A,B,C,D] : ( ( r2_wellord2(A,B) & r2_hidden(C,A) & r2_hidden(D,B) ) => r2_wellord2(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(B,k1_tarski(D))) ) ), file(card_1,t61_card_1), [interesting(0.00)]). fof(t144_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k9_relat_1(B,A),k2_relat_1(B)) ) ), file(relat_1,t144_relat_1), [interesting(0.00)]). fof(t84_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( v2_funct_1(B) => v2_funct_1(k7_relat_1(B,A)) ) ) ), file(funct_1,t84_funct_1), [interesting(0.00)]). fof(t91_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( r1_tarski(A,k1_relat_1(B)) => k1_relat_1(k7_relat_1(B,A)) = A ) ) ), file(relat_1,t91_relat_1), [interesting(0.00)]). fof(t148_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => k2_relat_1(k7_relat_1(B,A)) = k9_relat_1(B,A) ) ), file(relat_1,t148_relat_1), [interesting(0.00)]). fof(t58_card_3,theorem,( ! [A] : ( v1_finset_1(A) => r2_hidden(k1_card_1(A),k1_card_1(k5_ordinal2)) ) ), file(card_3,t58_card_3), [interesting(0.00)]). fof(t7_ordinal1,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & r1_tarski(B,A) ) ), file(ordinal1,t7_ordinal1), [interesting(0.00)]). fof(t26_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r1_ordinal1(A,B) | r2_hidden(B,A) ) ) ) ), file(ordinal1,t26_ordinal1), [interesting(0.00)]). fof(t16_card_4,theorem,( $true ), file(card_4,t16_card_4), [interesting(0.00)]). fof(t47_card_1,theorem,( k1_card_1(k1_xboole_0) = k1_xboole_0 ), file(card_1,t47_card_1), [interesting(0.00)]). fof(t59_card_2,theorem,( ! [A] : ( k1_card_1(A) = 0 => A = k1_xboole_0 ) ), file(card_2,t59_card_2), [interesting(0.00)]). fof(t18_card_4,theorem,( $true ), file(card_4,t18_card_4), [interesting(0.00)]). fof(t34_card_1,theorem,( ! [A,B] : ( k1_card_1(A) = k1_card_1(B) => k2_card_1(A) = k2_card_1(B) ) ), file(card_1,t34_card_1), [interesting(0.00)]). fof(t24_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ~ ( ~ r2_hidden(A,B) & A != B & ~ r2_hidden(B,A) ) ) ) ), file(ordinal1,t24_ordinal1), [interesting(0.00)]). fof(d6_card_1,definition,( ! [A,B] : ( v1_card_1(B) => ( B = k2_card_1(A) <=> ( r2_hidden(k1_card_1(A),B) & ! [C] : ( v1_card_1(C) => ( r2_hidden(k1_card_1(A),C) => r1_tarski(B,C) ) ) ) ) ) ), file(card_1,d6_card_1), [interesting(0.00)]). fof(t22_ordinal1,theorem,( ! [A] : ( v1_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( r1_tarski(A,B) & r2_hidden(B,C) ) => r2_hidden(A,C) ) ) ) ) ), file(ordinal1,t22_ordinal1), [interesting(0.00)]). fof(t32_card_1,theorem,( ! [A] : ( v1_card_1(A) => r2_hidden(A,k2_card_1(A)) ) ), file(card_1,t32_card_1), [interesting(0.00)]). fof(t76_card_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_card_1(k1_card_1(A)) = k1_card_1(k1_nat_1(A,1)) ) ), file(card_1,t76_card_1), [interesting(0.00)]). fof(t7_xboole_1,theorem,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), [interesting(0.00)]). fof(d8_xboole_0,definition,( ! [A,B] : ( r2_xboole_0(A,B) <=> ( r1_tarski(A,B) & A != B ) ) ), file(xboole_0,d8_xboole_0), [interesting(0.00)]). fof(t21_ordinal1,theorem,( ! [A] : ( v1_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r2_xboole_0(A,B) => r2_hidden(A,B) ) ) ) ), file(ordinal1,t21_ordinal1), [interesting(0.00)]). fof(t34_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r2_hidden(A,k1_ordinal1(B)) <=> r1_ordinal1(A,B) ) ) ) ), file(ordinal1,t34_ordinal1), [interesting(0.00)]). fof(t19_ordinal2,theorem, ( r2_hidden(k1_xboole_0,k5_ordinal2) & v4_ordinal1(k5_ordinal2) & ! [A] : ( v3_ordinal1(A) => ( ( r2_hidden(k1_xboole_0,A) & v4_ordinal1(A) ) => r1_ordinal1(k5_ordinal2,A) ) ) ), file(ordinal2,t19_ordinal2), [interesting(0.00)]). fof(t24_card_2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => k1_card_1(k14_ordinal2(A,B)) = k1_card_2(k1_card_1(A),k1_card_1(B)) ) ) ), file(card_2,t24_card_2), [interesting(0.00)]). fof(s3_ordinal2,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) & k1_relat_1(A) = f1_s3_ordinal2 & ! [B] : ( v3_ordinal1(B) => ( r2_hidden(B,f1_s3_ordinal2) => k1_funct_1(A,B) = f2_s3_ordinal2(B) ) ) ) ), file(ordinal2,s3_ordinal2), [interesting(0.00)]). fof(t46_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( v4_ordinal1(A) => ( A = k1_xboole_0 | ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) ) => ( ( k1_relat_1(C) = A & ! [D] : ( v3_ordinal1(D) => ( r2_hidden(D,A) => k1_funct_1(C,D) = k14_ordinal2(B,D) ) ) ) => k14_ordinal2(B,A) = k8_ordinal2(C) ) ) ) ) ) ) ), file(ordinal2,t46_ordinal2), [interesting(0.00)]). fof(t35_ordinal2,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) ) => ( k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A)) & k9_ordinal2(A) = k6_ordinal2(k2_relat_1(A)) ) ) ), file(ordinal2,t35_ordinal2), [interesting(0.00)]). fof(d1_zf_refle,definition,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r1_tarski(A,B) <=> ! [C] : ( v3_ordinal1(C) => ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ) ) ) ), file(zf_refle,d1_zf_refle), [interesting(0.00)]). fof(t29_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ~ ( r2_hidden(A,k7_ordinal2(B)) & ! [C] : ( v3_ordinal1(C) => ~ ( r2_hidden(C,B) & r1_ordinal1(A,C) ) ) ) ) ), file(ordinal2,t29_ordinal2), [interesting(0.00)]). fof(t23_ordinal1,theorem,( ! [A,B] : ( v3_ordinal1(B) => ( r2_hidden(A,B) => v3_ordinal1(A) ) ) ), file(ordinal1,t23_ordinal1), [interesting(0.00)]). fof(t65_card_1,theorem,( ! [A] : ( r2_hidden(A,k5_ordinal2) => v1_card_1(A) ) ), file(card_1,t65_card_1), [interesting(0.00)]). fof(d4_ordinal2,definition,( k4_ordinal2 = k1_ordinal1(k1_xboole_0) ), file(ordinal2,d4_ordinal2), [interesting(0.00)]). fof(t49_card_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_nat_1(A,B) = k14_ordinal2(A,B) ) ) ), file(card_2,t49_card_2), [interesting(0.00)]). fof(t41_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ( v4_ordinal1(A) <=> ! [B] : ( v3_ordinal1(B) => ( r2_hidden(B,A) => r2_hidden(k1_ordinal1(B),A) ) ) ) ) ), file(ordinal1,t41_ordinal1), [interesting(0.00)]). fof(t48_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => k14_ordinal2(A,k4_ordinal2) = k1_ordinal1(A) ) ), file(ordinal2,t48_ordinal2), [interesting(0.00)]). fof(t30_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ~ ( r1_ordinal1(A,B) & ! [C] : ( v3_ordinal1(C) => B != k14_ordinal2(A,C) ) ) ) ) ), file(ordinal3,t30_ordinal3), [interesting(0.00)]). fof(t27_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r1_ordinal1(A,k14_ordinal2(A,B)) & r1_ordinal1(B,k14_ordinal2(A,B)) ) ) ) ), file(ordinal3,t27_ordinal3), [interesting(0.00)]). fof(t33_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => k14_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(A,k14_ordinal2(B,C)) ) ) ) ), file(ordinal3,t33_ordinal3), [interesting(0.00)]). fof(d1_card_1,definition,( ! [A] : ( v1_card_1(A) <=> ? [B] : ( v3_ordinal1(B) & A = B & ! [C] : ( v3_ordinal1(C) => ( r2_wellord2(C,B) => r1_ordinal1(B,C) ) ) ) ) ), file(card_1,d1_card_1), [interesting(0.00)]). fof(t44_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => k14_ordinal2(A,k1_xboole_0) = A ) ), file(ordinal2,t44_ordinal2), [interesting(0.00)]). fof(t52_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => k15_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ), file(ordinal2,t52_ordinal2), [interesting(0.00)]). fof(t51_card_1,theorem,( 0 = k1_xboole_0 ), file(card_1,t51_card_1), [interesting(0.00)]). fof(t53_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => k15_ordinal2(k1_ordinal1(A),B) = k14_ordinal2(k15_ordinal2(A,B),B) ) ) ), file(ordinal2,t53_ordinal2), [interesting(0.00)]). fof(t45_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => k14_ordinal2(A,k1_ordinal1(B)) = k1_ordinal1(k14_ordinal2(A,B)) ) ) ), file(ordinal2,t45_ordinal2), [interesting(0.00)]). fof(t54_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( v4_ordinal1(A) => ( A = k1_xboole_0 | ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) ) => ( ( k1_relat_1(C) = A & ! [D] : ( v3_ordinal1(D) => ( r2_hidden(D,A) => k1_funct_1(C,D) = k15_ordinal2(D,B) ) ) ) => k15_ordinal2(A,B) = k3_tarski(k8_ordinal2(C)) ) ) ) ) ) ) ), file(ordinal2,t54_ordinal2), [interesting(0.00)]). fof(d4_tarski,definition,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), [interesting(0.00)]). fof(t19_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v1_ordinal1(C) => ( ( r2_hidden(C,A) & r2_hidden(A,B) ) => r2_hidden(C,B) ) ) ) ) ), file(ordinal1,t19_ordinal1), [interesting(0.00)]). fof(d2_ordinal1,definition,( ! [A] : ( v1_ordinal1(A) <=> ! [B] : ( r2_hidden(B,A) => r1_tarski(B,A) ) ) ), file(ordinal1,d2_ordinal1), [interesting(0.00)]). fof(t56_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ( k15_ordinal2(k4_ordinal2,A) = A & k15_ordinal2(A,k4_ordinal2) = A ) ) ), file(ordinal2,t56_ordinal2), [interesting(0.00)]). fof(t59_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( r1_ordinal1(A,B) => r1_ordinal1(k15_ordinal2(C,A),k15_ordinal2(C,B)) ) ) ) ) ), file(ordinal2,t59_ordinal2), [interesting(0.00)]). fof(s1_ordinal2,theorem, ( ( p1_s1_ordinal2(k1_xboole_0) & ! [A] : ( v3_ordinal1(A) => ( p1_s1_ordinal2(A) => p1_s1_ordinal2(k1_ordinal1(A)) ) ) & ! [A] : ( v3_ordinal1(A) => ( ( v4_ordinal1(A) & ! [B] : ( v3_ordinal1(B) => ( r2_hidden(B,A) => p1_s1_ordinal2(B) ) ) ) => ( A = k1_xboole_0 | p1_s1_ordinal2(A) ) ) ) ) => ! [A] : ( v3_ordinal1(A) => p1_s1_ordinal2(A) ) ), file(ordinal2,s1_ordinal2), [interesting(0.00)]). fof(t48_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( v4_ordinal1(A) => v4_ordinal1(k15_ordinal2(A,B)) ) ) ) ), file(ordinal3,t48_ordinal3), [interesting(0.00)]). fof(t25_card_2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => k1_card_1(k15_ordinal2(A,B)) = k2_card_2(k1_card_1(A),k1_card_1(B)) ) ) ), file(card_2,t25_card_2), [interesting(0.00)]). fof(d1_card_2,definition,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => k1_card_2(A,B) = k1_card_1(k14_ordinal2(A,B)) ) ) ), file(card_2,d1_card_2), [interesting(0.00)]). fof(t47_card_2,theorem,( ! [A,B] : r1_tarski(k1_card_1(k2_xboole_0(A,B)),k1_card_2(k1_card_1(A),k1_card_1(B))) ), file(card_2,t47_card_2), [interesting(0.00)]). fof(t13_card_1,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(A,B) <=> ( r1_tarski(A,B) & A != B ) ) ) ) ), file(card_1,t13_card_1), [interesting(0.00)]). fof(t50_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( r1_ordinal1(A,B) => r1_ordinal1(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) ), file(ordinal2,t50_ordinal2), [interesting(0.00)]). fof(t51_card_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_card_1(k1_nat_1(A,B)) = k1_card_2(k1_card_1(A),k1_card_1(B)) ) ) ), file(card_2,t51_card_2), [interesting(0.00)]). fof(t25_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => r3_xboole_0(A,B) ) ) ), file(ordinal1,t25_ordinal1), [interesting(0.00)]). fof(d9_xboole_0,definition,( ! [A,B] : ( r3_xboole_0(A,B) <=> ( r1_tarski(A,B) | r1_tarski(B,A) ) ) ), file(xboole_0,d9_xboole_0), [interesting(0.00)]). fof(t13_finset_1,theorem,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), [interesting(0.00)]). fof(t52_card_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_card_1(k2_nat_1(A,B)) = k2_card_2(k1_card_1(A),k1_card_1(B)) ) ) ), file(card_2,t52_card_2), [interesting(0.00)]). fof(t21_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ! [D] : ( v3_ordinal1(D) => ( ( r1_ordinal1(A,B) & r1_ordinal1(C,D) ) => r1_ordinal1(k14_ordinal2(A,C),k14_ordinal2(B,D)) ) ) ) ) ) ), file(ordinal3,t21_ordinal3), [interesting(0.00)]). fof(t41_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ! [D] : ( v1_card_1(D) => ( ~ ( ~ ( r2_hidden(A,B) & r2_hidden(C,D) ) & ~ ( r1_tarski(A,B) & r2_hidden(C,D) ) & ~ ( r2_hidden(A,B) & r1_tarski(C,D) ) & ~ ( r1_tarski(A,B) & r1_tarski(C,D) ) ) => ( r1_tarski(k1_card_2(A,C),k1_card_2(B,D)) & r1_tarski(k1_card_2(C,A),k1_card_2(B,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_card_1,d1_card_2,t21_ordinal3,t27_card_1,t27_card_1]), [file(card_4,t41_card_4),interesting(0.00)]). fof(t42_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ( ( r2_hidden(A,B) | r1_tarski(A,B) ) => ( r1_tarski(k1_card_2(C,A),k1_card_2(C,B)) & r1_tarski(k1_card_2(C,A),k1_card_2(B,C)) & r1_tarski(k1_card_2(A,C),k1_card_2(C,B)) & r1_tarski(k1_card_2(A,C),k1_card_2(B,C)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t41_card_4]), [file(card_4,t42_card_4),interesting(0.00)]). fof(t17_xboole_1,theorem,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_xboole_1), [interesting(0.00)]). fof(t1_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), [interesting(0.00)]). fof(d11_card_1,definition,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k4_card_1(A) <=> k1_card_1(B) = k1_card_1(A) ) ) ) ), file(card_1,d11_card_1), [interesting(0.00)]). fof(t36_xboole_1,theorem,( ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ), file(xboole_1,t36_xboole_1), [interesting(0.00)]). fof(d2_card_2,definition,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => k2_card_2(A,B) = k1_card_1(k2_zfmisc_1(A,B)) ) ) ), file(card_2,d2_card_2), [interesting(0.00)]). fof(t71_relat_1,theorem,( ! [A] : ( k1_relat_1(k6_relat_1(A)) = A & k2_relat_1(k6_relat_1(A)) = A ) ), file(relat_1,t71_relat_1), [interesting(0.00)]). fof(t35_funct_1,theorem,( ! [A,B] : ( r2_hidden(B,A) => k1_funct_1(k6_relat_1(A),B) = B ) ), file(funct_1,t35_funct_1), [interesting(0.00)]). fof(t54_card_3,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => r1_tarski(k1_card_1(k3_card_3(A)),k6_card_3(k1_card_3(A))) ) ), file(card_3,t54_card_3), [interesting(0.00)]). fof(d2_card_3,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_card_3(B) ) => ( B = k1_card_3(A) <=> ( k1_relat_1(B) = k1_relat_1(A) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(B,C) = k1_card_1(k1_funct_1(A,C)) ) ) ) ) ) ), file(card_3,d2_card_3), [interesting(0.00)]). fof(t19_funcop_1,theorem,( ! [A,B] : ( k1_relat_1(k2_funcop_1(A,B)) = A & r1_tarski(k2_relat_1(k2_funcop_1(A,B)),k1_tarski(B)) ) ), file(funcop_1,t19_funcop_1), [interesting(0.00)]). fof(t13_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(B,A) => k1_funct_1(k2_funcop_1(A,C),B) = C ) ), file(funcop_1,t13_funcop_1), [interesting(0.00)]). fof(t43_card_3,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_card_3(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_card_3(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => r1_tarski(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) => r1_tarski(k6_card_3(A),k6_card_3(B)) ) ) ) ), file(card_3,t43_card_3), [interesting(0.00)]). fof(d7_card_3,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_card_3(A) ) => k6_card_3(A) = k1_card_1(k3_card_3(k2_card_3(A))) ) ), file(card_3,d7_card_3), [interesting(0.00)]). fof(t36_card_3,theorem,( ! [A,B] : k3_card_3(k2_card_3(k2_funcop_1(A,B))) = k2_zfmisc_1(B,A) ), file(card_3,t36_card_3), [interesting(0.00)]). fof(t14_card_2,theorem,( ! [A,B] : ( r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(k1_card_1(A),B)) & r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(A,k1_card_1(B))) & r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(k1_card_1(A),k1_card_1(B))) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(k1_card_1(A),B)) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(A,k1_card_1(B))) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(k1_card_1(A),k1_card_1(B))) ) ), file(card_2,t14_card_2), [interesting(0.00)]). fof(t11_card_2,theorem,( ! [A,B] : ( r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(B,A)) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(B,A)) ) ), file(card_2,t11_card_2), [interesting(0.00)]). fof(d4_card_3,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => k3_card_3(A) = k3_tarski(k2_relat_1(A)) ) ), file(card_3,d4_card_3), [interesting(0.00)]). fof(s3_card_3,theorem, ( ( f1_s3_card_3 != k1_xboole_0 & ! [A,B] : ( p1_s3_card_3(A,B) | p1_s3_card_3(B,A) ) & ! [A,B,C] : ( ( p1_s3_card_3(A,B) & p1_s3_card_3(B,C) ) => p1_s3_card_3(A,C) ) ) => ? [A] : ( r2_hidden(A,f1_s3_card_3) & ! [B] : ( r2_hidden(B,f1_s3_card_3) => p1_s3_card_3(A,B) ) ) ), file(card_3,s3_card_3), [interesting(0.00)]). fof(t14_card_3,theorem,( k3_card_3(k1_xboole_0) = k1_xboole_0 ), file(card_3,t14_card_3), [interesting(0.00)]). fof(t64_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ( ( k1_relat_1(A) = k1_xboole_0 | k2_relat_1(A) = k1_xboole_0 ) => A = k1_xboole_0 ) ) ), file(relat_1,t64_relat_1), [interesting(0.00)]). fof(t64_card_4,theorem,( $true ), file(card_4,t64_card_4), [interesting(0.00)]). fof(s2_funct_1,theorem, ( ( ! [A,B,C] : ( ( r2_hidden(A,f1_s2_funct_1) & p1_s2_funct_1(A,B) & p1_s2_funct_1(A,C) ) => B = C ) & ! [A] : ~ ( r2_hidden(A,f1_s2_funct_1) & ! [B] : ~ p1_s2_funct_1(A,B) ) ) => ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s2_funct_1 & ! [B] : ( r2_hidden(B,f1_s2_funct_1) => p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) ), file(funct_1,s2_funct_1), [interesting(0.00)]). fof(d8_finseq_1,definition,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( B = k9_finseq_1(A) <=> ( k1_relat_1(B) = k2_finseq_1(1) & k1_funct_1(B,1) = A ) ) ) ), file(finseq_1,d8_finseq_1), [interesting(0.00)]). fof(t116_finseq_2,theorem,( ! [A] : ( ~ v1_xboole_0(A) => k4_finseq_2(1,A) = a_1_0_finseq_2(A) ) ), file(finseq_2,t116_finseq_2), [interesting(0.00)]). fof(t117_finseq_2,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_2(B,A,k4_finseq_2(1,A)) => ? [C] : ( m1_subset_1(C,A) & B = k12_finseq_1(A,C) ) ) ) ), file(finseq_2,t117_finseq_2), [interesting(0.00)]). fof(t112_finseq_2,theorem,( ! [A] : k4_finseq_2(0,A) = k6_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,A)),k6_finseq_1(A)) ), file(finseq_2,t112_finseq_2), [interesting(0.00)]). fof(t119_zfmisc_1,theorem,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) ) => r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ), file(zfmisc_1,t119_zfmisc_1), [interesting(0.00)]). fof(t33_zfmisc_1,theorem,( ! [A,B,C,D] : ( k4_tarski(A,B) = k4_tarski(C,D) => ( A = C & B = D ) ) ), file(zfmisc_1,t33_zfmisc_1), [interesting(0.00)]). fof(t109_finseq_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_finseq_2(C,B,k4_finseq_2(A,B)) => k3_finseq_1(C) = A ) ) ) ), file(finseq_2,t109_finseq_2), [interesting(0.00)]). fof(t64_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v1_finseq_1(D) ) => ~ ( k7_finseq_1(A,B) = k7_finseq_1(C,D) & r1_xreal_0(k3_finseq_1(A),k3_finseq_1(C)) & ! [E] : ( ( v1_relat_1(E) & v1_funct_1(E) & v1_finseq_1(E) ) => k7_finseq_1(A,E) != C ) ) ) ) ) ) ), file(finseq_1,t64_finseq_1), [interesting(0.00)]). fof(t35_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ), file(finseq_1,t35_finseq_1), [interesting(0.00)]). fof(t23_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( k2_xcmplx_0(A,B) = 0 => ( A = 0 & B = 0 ) ) ) ) ), file(nat_1,t23_nat_1), [interesting(0.00)]). fof(t25_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k3_finseq_1(A) = 0 <=> A = k1_xboole_0 ) ) ), file(finseq_1,t25_finseq_1), [interesting(0.00)]). fof(t47_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k7_finseq_1(A,k1_xboole_0) = A & k7_finseq_1(k1_xboole_0,A) = A ) ) ), file(finseq_1,t47_finseq_1), [interesting(0.00)]). fof(t46_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( k7_finseq_1(A,B) = k7_finseq_1(C,B) | k7_finseq_1(B,A) = k7_finseq_1(B,C) ) => A = C ) ) ) ) ), file(finseq_1,t46_finseq_1), [interesting(0.00)]). fof(t127_finseq_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m2_finseq_2(D,C,k4_finseq_2(A,C)) => ! [E] : ( m2_finseq_2(E,C,k4_finseq_2(B,C)) => m2_finseq_2(k8_finseq_1(C,D,E),C,k4_finseq_2(k1_nat_1(A,B),C)) ) ) ) ) ) ), file(finseq_2,t127_finseq_2), [interesting(0.00)]). fof(t126_finseq_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m2_finseq_2(D,C,k4_finseq_2(k1_nat_1(A,B),C)) => ? [E] : ( m2_finseq_2(E,C,k4_finseq_2(A,C)) & ? [F] : ( m2_finseq_2(F,C,k4_finseq_2(B,C)) & D = k8_finseq_1(C,E,F) ) ) ) ) ) ) ), file(finseq_2,t126_finseq_2), [interesting(0.00)]). fof(t128_finseq_2,theorem,( ! [A] : ( ~ v1_xboole_0(A) => k3_finseq_2(A) = k3_tarski(a_1_3_finseq_2(A)) ) ), file(finseq_2,t128_finseq_2), [interesting(0.00)]). fof(t66_finseq_1,theorem,( ! [A] : r2_hidden(k1_xboole_0,k13_finseq_1(A)) ), file(finseq_1,t66_finseq_1), [interesting(0.00)]). fof(d12_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k9_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( r2_hidden(E,k1_relat_1(A)) & r2_hidden(E,B) & D = k1_funct_1(A,E) ) ) ) ) ), file(funct_1,d12_funct_1), [interesting(0.00)]). fof(d11_finseq_1,definition,( ! [A,B] : ( B = k13_finseq_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> m2_finseq_1(C,A) ) ) ), file(finseq_1,d11_finseq_1), [interesting(0.00)]). fof(t57_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k1_funct_1(B,1) = A ) ) ) ), file(finseq_1,t57_finseq_1), [interesting(0.00)]). fof(t2_card_2,theorem,( ! [A,B] : ( r1_tarski(k1_card_1(A),k1_card_1(B)) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & r1_tarski(A,k9_relat_1(C,B)) ) ) ), file(card_2,t2_card_2), [interesting(0.00)]). fof(t68_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ! [D] : ( v1_card_1(D) => ( ~ ( ~ ( r2_hidden(A,B) & r2_hidden(C,D) ) & ~ ( r1_tarski(A,B) & r2_hidden(C,D) ) & ~ ( r2_hidden(A,B) & r1_tarski(C,D) ) & ~ ( r1_tarski(A,B) & r1_tarski(C,D) ) ) => ( r1_tarski(k2_card_2(A,C),k2_card_2(B,D)) & r1_tarski(k2_card_2(C,A),k2_card_2(B,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_card_1,d2_card_2,t119_zfmisc_1,t27_card_1,t27_card_1]), [file(card_4,t68_card_4),interesting(0.00)]). fof(t69_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ( ( r2_hidden(A,B) | r1_tarski(A,B) ) => ( r1_tarski(k2_card_2(C,A),k2_card_2(C,B)) & r1_tarski(k2_card_2(C,A),k2_card_2(B,C)) & r1_tarski(k2_card_2(A,C),k2_card_2(C,B)) & r1_tarski(k2_card_2(A,C),k2_card_2(B,C)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t68_card_4]), [file(card_4,t69_card_4),interesting(0.00)]). fof(t11_funct_2,theorem,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => r2_hidden(C,k1_funct_2(A,B)) ) ) ), file(funct_2,t11_funct_2), [interesting(0.00)]). fof(t2_xboole_1,theorem,( ! [A] : r1_tarski(k1_xboole_0,A) ), file(xboole_1,t2_xboole_1), [interesting(0.00)]). fof(t79_xboole_1,theorem,( ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) ), file(xboole_1,t79_xboole_1), [interesting(0.00)]). fof(d3_card_2,definition,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => k3_card_2(A,B) = k1_card_1(k1_funct_2(B,A)) ) ) ), file(card_2,d3_card_2), [interesting(0.00)]). fof(t63_funct_5,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k1_funct_2(C,A),k1_funct_2(C,B)) ) ), file(funct_5,t63_funct_5), [interesting(0.00)]). fof(d7_xboole_0,definition,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), [interesting(0.00)]). fof(t45_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => B = k2_xboole_0(A,k4_xboole_0(B,A)) ) ), file(xboole_1,t45_xboole_1), [interesting(0.00)]). fof(t69_funct_5,theorem,( ! [A,B,C] : ( r1_xboole_0(A,B) => ( r2_wellord2(k1_funct_2(k2_xboole_0(A,B),C),k2_zfmisc_1(k1_funct_2(A,C),k1_funct_2(B,C))) & k1_card_1(k1_funct_2(k2_xboole_0(A,B),C)) = k1_card_1(k2_zfmisc_1(k1_funct_2(A,C),k1_funct_2(B,C))) ) ) ), file(funct_5,t69_funct_5), [interesting(0.00)]). fof(t33_card_2,theorem,( ! [A] : ( v1_card_1(A) => k2_card_2(A,1) = A ) ), file(card_2,t33_card_2), [interesting(0.00)]). fof(t70_card_4,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => ! [D] : ( v1_card_1(D) => ~ ( ~ ( ~ ( r2_hidden(A,B) & r2_hidden(C,D) ) & ~ ( r1_tarski(A,B) & r2_hidden(C,D) ) & ~ ( r2_hidden(A,B) & r1_tarski(C,D) ) & ~ ( r1_tarski(A,B) & r1_tarski(C,D) ) ) & A != 0 & ~ r1_tarski(k3_card_2(A,C),k3_card_2(B,D)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_card_1,t11_funct_2,t46_card_1,t2_xboole_1,t21_card_1,t47_card_1,t13_card_1,d5_card_1,d6_card_1,l1_card_4,t76_card_1,t79_xboole_1,d3_card_2,t63_funct_5,d7_xboole_0,t45_xboole_1,t27_card_1,t14_card_2,d2_card_2,t69_funct_5,t68_card_4,t33_card_2,t1_xboole_1,t13_card_1]), [file(card_4,t70_card_4),interesting(0.00)]). fof(t39_card_2,theorem,( ! [A] : ( v1_card_1(A) => ( A != 0 => k3_card_2(0,A) = 0 ) ) ), file(card_2,t39_card_2), [interesting(0.00)]). fof(t39_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( A != k1_xboole_0 => ( r1_ordinal1(B,k15_ordinal2(B,A)) & r1_ordinal1(B,k15_ordinal2(A,B)) ) ) ) ) ), file(ordinal3,t39_ordinal3), [interesting(0.00)]). fof(s1_xboole_0,theorem,( ? [A] : ! [B] : ( r2_hidden(B,A) <=> ( r2_hidden(B,f1_s1_xboole_0) & p1_s1_xboole_0(B) ) ) ), file(xboole_0,s1_xboole_0), [interesting(0.00)]). fof(d5_partfun1,definition,( ! [A,B,C] : ( C = k4_partfun1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( v1_relat_1(E) & v1_funct_1(E) & D = E & r1_tarski(k1_relat_1(E),A) & r1_tarski(k2_relat_1(E),B) ) ) ) ), file(partfun1,d5_partfun1), [interesting(0.00)]). fof(d1_relat_1,definition,( ! [A] : ( v1_relat_1(A) <=> ! [B] : ~ ( r2_hidden(B,A) & ! [C,D] : B != k4_tarski(C,D) ) ) ), file(relat_1,d1_relat_1), [interesting(0.00)]). fof(d1_funct_1,definition,( ! [A] : ( v1_funct_1(A) <=> ! [B,C,D] : ( ( r2_hidden(k4_tarski(B,C),A) & r2_hidden(k4_tarski(B,D),A) ) => C = D ) ) ), file(funct_1,d1_funct_1), [interesting(0.00)]). fof(d9_ordinal1,definition,( ! [A] : ( v6_ordinal1(A) <=> ! [B,C] : ( ( r2_hidden(B,A) & r2_hidden(C,A) ) => r3_xboole_0(B,C) ) ) ), file(ordinal1,d9_ordinal1), [interesting(0.00)]). fof(d4_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) => ( C = k1_funct_1(A,B) <=> r2_hidden(k4_tarski(B,C),A) ) ) & ( ~ r2_hidden(B,k1_relat_1(A)) => ( C = k1_funct_1(A,B) <=> C = k1_xboole_0 ) ) ) ) ), file(funct_1,d4_funct_1), [interesting(0.00)]). fof(t8_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(k4_tarski(A,B),C) <=> ( r2_hidden(A,k1_relat_1(C)) & B = k1_funct_1(C,A) ) ) ) ), file(funct_1,t8_funct_1), [interesting(0.00)]). fof(t92_zfmisc_1,theorem,( ! [A,B] : ( r2_hidden(A,B) => r1_tarski(A,k3_tarski(B)) ) ), file(zfmisc_1,t92_zfmisc_1), [interesting(0.00)]). fof(t25_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(A,B) => ( r1_tarski(k1_relat_1(A),k1_relat_1(B)) & r1_tarski(k2_relat_1(A),k2_relat_1(B)) ) ) ) ) ), file(relat_1,t25_relat_1), [interesting(0.00)]). fof(t106_zfmisc_1,theorem,( ! [A,B,C,D] : ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D)) <=> ( r2_hidden(A,C) & r2_hidden(B,D) ) ) ), file(zfmisc_1,t106_zfmisc_1), [interesting(0.00)]). fof(t177_orders_1,theorem,( ! [A] : ~ ( A != k1_xboole_0 & ! [B] : ( ( r1_tarski(B,A) & v6_ordinal1(B) ) => ( B = k1_xboole_0 | r2_hidden(k3_tarski(B),A) ) ) & ! [B] : ~ ( r2_hidden(B,A) & ! [C] : ~ ( r2_hidden(C,A) & C != B & r1_tarski(B,C) ) ) ) ), file(orders_1,t177_orders_1), [interesting(0.00)]). fof(t39_xboole_1,theorem,( ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ), file(xboole_1,t39_xboole_1), [interesting(0.00)]). fof(t48_card_2,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => k1_card_1(k2_xboole_0(A,B)) = k1_card_2(k1_card_1(A),k1_card_1(B)) ) ), file(card_2,t48_card_2), [interesting(0.00)]). fof(t73_card_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(k1_card_1(A),k1_card_1(B)) <=> ~ r1_xreal_0(B,A) ) ) ) ), file(card_1,t73_card_1), [interesting(0.00)]). fof(t10_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(C,D) & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ), file(xreal_1,t10_xreal_1), [interesting(0.00)]). fof(t71_card_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( k1_card_1(A) = k1_card_1(B) => A = B ) ) ) ), file(card_1,t71_card_1), [interesting(0.00)]). fof(t113_zfmisc_1,theorem,( ! [A,B] : ( k2_zfmisc_1(A,B) = k1_xboole_0 <=> ( A = k1_xboole_0 | B = k1_xboole_0 ) ) ), file(zfmisc_1,t113_zfmisc_1), [interesting(0.00)]). fof(t123_zfmisc_1,theorem,( ! [A,B,C,D] : k2_zfmisc_1(k3_xboole_0(A,B),k3_xboole_0(C,D)) = k3_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ), file(zfmisc_1,t123_zfmisc_1), [interesting(0.00)]). fof(t23_xboole_1,theorem,( ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ), file(xboole_1,t23_xboole_1), [interesting(0.00)]). fof(t32_card_2,theorem,( ! [A] : ( v1_card_1(A) => k2_card_2(A,0) = 0 ) ), file(card_2,t32_card_2), [interesting(0.00)]). fof(t18_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(k2_relat_1(k1_funct_4(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ), file(funct_4,t18_funct_4), [interesting(0.00)]). fof(t19_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(k2_relat_1(A),k2_relat_1(k1_funct_4(B,A))) ) ) ), file(funct_4,t19_funct_4), [interesting(0.00)]). fof(t33_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B)) => r1_tarski(A,k1_funct_4(A,B)) ) ) ) ), file(funct_4,t33_funct_4), [interesting(0.00)]). fof(d1_funct_4,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( C = k1_funct_4(A,B) <=> ( k1_relat_1(C) = k2_xboole_0(k1_relat_1(A),k1_relat_1(B)) & ! [D] : ( r2_hidden(D,k2_xboole_0(k1_relat_1(A),k1_relat_1(B))) => ( ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(C,D) = k1_funct_1(B,D) ) & ( ~ r2_hidden(D,k1_relat_1(B)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ), file(funct_4,d1_funct_4), [interesting(0.00)]). fof(t8_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), [interesting(0.00)]). fof(t120_zfmisc_1,theorem,( ! [A,B,C] : ( k2_zfmisc_1(k2_xboole_0(A,B),C) = k2_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C)) & k2_zfmisc_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ), file(zfmisc_1,t120_zfmisc_1), [interesting(0.00)]). fof(t4_xboole_1,theorem,( ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ), file(xboole_1,t4_xboole_1), [interesting(0.00)]). fof(d2_xboole_0,definition,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), [interesting(0.00)]). fof(t14_funct_4,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) ), file(funct_4,t14_funct_4), [interesting(0.00)]). fof(t17_funct_4,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( r1_xboole_0(k1_relat_1(B),k1_relat_1(C)) & r2_hidden(A,k1_relat_1(B)) ) => k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) ), file(funct_4,t17_funct_4), [interesting(0.00)]). fof(d3_xboole_0,definition,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), [interesting(0.00)]). fof(t26_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(A,k1_funct_4(B,A)) ) ) ), file(funct_4,t26_funct_4), [interesting(0.00)]). fof(t4_card_2,theorem,( ! [A,B] : ~ ( r2_hidden(k1_card_1(A),k1_card_1(B)) & k4_xboole_0(B,A) = k1_xboole_0 ) ), file(card_2,t4_card_2), [interesting(0.00)]). fof(t26_xboole_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) ), file(xboole_1,t26_xboole_1), [interesting(0.00)]). fof(t3_xboole_1,theorem,( ! [A] : ( r1_tarski(A,k1_xboole_0) => A = k1_xboole_0 ) ), file(xboole_1,t3_xboole_1), [interesting(0.00)]). fof(t40_xboole_1,theorem,( ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),B) = k4_xboole_0(A,B) ), file(xboole_1,t40_xboole_1), [interesting(0.00)]). fof(t83_xboole_1,theorem,( ! [A,B] : ( r1_xboole_0(A,B) <=> k4_xboole_0(A,B) = A ) ), file(xboole_1,t83_xboole_1), [interesting(0.00)]). fof(t126_zfmisc_1,theorem,( ! [A,B,C,D] : k4_xboole_0(k2_zfmisc_1(A,B),k2_zfmisc_1(C,D)) = k2_xboole_0(k2_zfmisc_1(k4_xboole_0(A,C),B),k2_zfmisc_1(A,k4_xboole_0(B,D))) ), file(zfmisc_1,t126_zfmisc_1), [interesting(0.00)]). fof(t12_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k2_xboole_0(A,B) = B ) ), file(xboole_1,t12_xboole_1), [interesting(0.00)]). fof(t66_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(0,C) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t66_xreal_1), [interesting(0.00)]). fof(t70_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(C,B) & r1_xreal_0(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ), file(xreal_1,t70_xreal_1), [interesting(0.00)]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.00)]).