fof(t44_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => k3_card_2(k2_card_1(A),B) = k2_card_2(k3_card_2(A,B),k2_card_1(A)) ) ) ), inference(mizar_proof,[status(thm)],[t23_card_5,t13_card_1,t22_card_4,t43_card_5,t22_ordinal1,t27_card_5,t35_card_5,t23_card_4,s2_ordinal2,d3_tarski,d2_funct_2,d7_ordinal1,d8_ordinal2,t39_card_5,t20_zfrefle1,d5_funct_1,d2_funct_2,d4_tarski,t27_card_1,d3_card_2,t54_card_3,d2_card_3,t19_funcop_1,t1_xboole_1,t23_ordinal1,d2_card_3,t13_funcop_1,t24_card_1,d5_card_1,t22_ordinal1,t23_card_4,t63_funct_5,t27_card_1,d3_card_2,t68_funct_5,t43_card_3,t52_card_3,t1_xboole_1,l1_card_5,t25_card_5,t32_card_1,t40_card_2,t70_card_4,t77_card_4,t68_card_4,d10_xboole_0,t14_card_1]), [file(card_5,t44_card_5),interesting(1.00)]). fof(t40_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ( r2_hidden(k1_card_5(A),A) => v2_card_1(A) ) ) ), inference(mizar_proof,[status(thm)],[d7_card_1,t23_card_4,t26_card_5,t35_card_5]), [file(card_5,t40_card_5),interesting(0.99)]). fof(t47_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => ( ( r1_tarski(k1_card_5(A),B) & r2_hidden(B,A) ) => k3_card_2(A,B) = k3_card_2(k6_card_3(k2_card_5(A,B)),k1_card_5(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[t45_card_5,t27_card_5,t43_card_2,t71_card_4,t22_ordinal1,t41_card_5,d5_card_1,d3_card_2,d2_funct_2,s1_partfun1,s1_zfrefle1,s1_zfrefle1,d8_funct_1,d5_funct_1,t32_card_5,d6_ordinal1,t15_ordinal3,t41_ordinal1,t29_ordinal2,d5_funct_1,t10_ordinal1,t7_xboole_1,t22_ordinal1,t33_zfmisc_1,t9_funct_1,s3_funct_1,d3_tarski,d5_funct_1,d3_tarski,d5_funct_1,d3_tarski,d5_funct_1,t23_ordinal1,d5_funct_1,t10_ordinal3,d2_funct_2,d5_funct_1,d1_card_3,d3_card_5,t7_mcart_1,t33_card_3,d2_funct_2,t26_card_1,d5_card_1,t68_funct_5,d3_card_2,d2_card_3,d3_card_3,d2_card_3,d3_card_3,t13_card_2,t9_funct_1,d2_card_3,d3_card_5,d3_card_5,d3_card_5,d3_card_2,t9_funct_1,t54_card_3,t71_card_4,t1_xboole_1,d10_xboole_0]), [file(card_5,t47_card_5),interesting(0.92)]). fof(t35_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => k1_card_5(k2_card_1(A)) = k2_card_1(A) ) ), inference(mizar_proof,[status(thm)],[d2_card_5,d5_zfrefle1,d2_card_3,t19_funcop_1,d2_card_3,t13_funcop_1,d5_funct_1,t23_ordinal1,t24_card_1,t22_ordinal1,t23_card_4,t43_card_3,t52_card_3,t54_card_3,d8_ordinal2,t8_ordinal3,t1_xboole_1,t27_zf_refle,t27_card_1,d5_card_1,t24_card_2,t42_card_4,t48_ordinal2,d2_card_5,t19_card_2,t1_xboole_1,t25_ordinal1,l1_card_5,t25_card_5,t29_card_2,t32_card_2,t14_card_1,t19_card_2,t2_xboole_1,l1_card_5,t25_card_5,t13_card_1,d9_xboole_0,t26_card_5,t27_card_5,t32_card_1,t34_card_4,t78_card_4,l1_card_5,t25_card_5,t14_card_1,t1_xboole_1,t34_card_4,d10_xboole_0]), [file(card_5,t35_card_5),interesting(0.91)]). fof(t17_card_5,theorem,( ! [A] : ( v3_ordinal1(A) => ? [B] : ( v3_ordinal1(B) & r1_ordinal1(B,k1_card_1(A)) & r4_zfrefle1(A,B) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,d4_wellord2,s1_xboole_0,d7_ordinal1,d8_ordinal2,d3_tarski,t9_wellord2,d2_wellord2,t50_wellord1,d9_wellord1,d1_wellord2,d7_wellord1,d7_ordinal1,d8_ordinal2,t17_wellord2,t45_relat_1,t46_relat_1,d7_ordinal1,d8_ordinal2,d5_zfrefle1,t45_relat_1,t46_relat_1,d16_ordinal2,t46_funct_1,t19_ordinal1,d2_ordinal1,d5_funct_1,d8_funct_1,d1_wellord2,d7_wellord1,d1_wellord2,d8_xboole_0,d8_funct_1,t21_ordinal1,t73_funct_1,t26_ordinal1,t92_zfmisc_1,t22_funct_1,t1_xboole_1,d8_xboole_0,t21_ordinal1,d3_tarski,d5_funct_1,t23_ordinal1,d5_funct_1,t22_funct_1,d5_funct_1,t27_ordinal2,t10_card_5,t23_ordinal1,t90_relat_1,t70_funct_1,d3_xboole_0,s1_ordinal1,t19_ordinal1,t10_card_5,t23_ordinal1,t70_funct_1,t90_relat_1,t19_ordinal1,d3_xboole_0,t7_ordinal1,t19_ordinal1,d5_funct_1,t23_ordinal1,t22_funct_1,d5_funct_1,t27_ordinal2,t19_ordinal1,d10_xboole_0,t26_ordinal2,t30_ordinal2]), [file(card_5,t17_card_5),interesting(0.91)]). fof(t12_card_5,theorem,( ! [A] : ( v1_card_1(A) => ~ ( ~ v1_finset_1(A) & ! [B] : ( v3_ordinal1(B) => A != k3_card_1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t76_card_1,t39_card_1,s1_xboole_0,t42_card_1,s2_funct_1,d5_funct_1,d3_tarski,t23_ordinal1,t41_card_1,t11_card_5,t19_ordinal1,t42_card_1,d5_funct_1,t31_ordinal1,s2_ordinal2,d5_funct_1,t39_card_1,d7_card_1,t22_card_4,t11_card_5,t13_card_1,t42_card_1,d5_funct_1,t41_ordinal1,t40_card_1,d3_tarski,t23_ordinal1,t29_ordinal2,d5_funct_1,t23_ordinal1,t11_card_5,d5_funct_1,t22_ordinal1,t27_card_1,d5_card_1,t11_card_5,t65_relat_1,t14_card_1,t11_card_4,d10_xboole_0,t11_card_5,t13_card_1,t42_card_1,d5_funct_1,s1_card_1]), [file(card_5,t12_card_5),interesting(0.89)]). fof(t22_card_5,theorem,( ! [A,B] : ( v1_card_1(B) => ( A != k1_xboole_0 => r1_tarski(k1_card_1(a_2_0_card_5(A,B)),k2_card_2(B,k3_card_2(k1_card_1(A),B))) ) ) ), inference(mizar_proof,[status(thm)],[t46_card_1,d5_card_1,t21_card_1,d5_card_1,d4_wellord2,t19_card_5,t20_card_5,d4_wellord2,t21_card_1,t3_card_2,t22_ordinal1,t60_card_3,t91_relat_1,t97_relat_1,t23_ordinal1,t26_ordinal1,t72_funct_1,t19_card_5,t9_funct_1,t144_relat_1,t16_card_5,t3_card_2,t22_ordinal1,t14_card_5,s3_ordinal2,t60_card_3,d2_ordinal1,t21_card_5,d6_ordinal3,d1_ordinal4,t66_ordinal3,d6_ordinal3,d1_ordinal4,s2_funct_1,d8_funct_1,t33_zfmisc_1,d3_tarski,d12_funct_1,d12_funct_1,d8_funct_1,d10_xboole_0,d3_tarski,d12_funct_1,d12_funct_1,d8_funct_1,d4_wellord2,d3_tarski,d5_funct_1,t20_card_5,d4_wellord2,t21_card_1,t3_card_2,t22_ordinal1,t60_card_3,t91_relat_1,t97_relat_1,d3_tarski,d5_funct_1,t23_ordinal1,t26_ordinal1,t73_funct_1,t144_relat_1,t10_ordinal3,d2_funct_2,t106_zfmisc_1,t27_card_1,t14_card_2,d2_card_2,d3_card_2,t21_card_1]), [file(card_5,t22_card_5),interesting(0.87)]). fof(t38_card_5,theorem,( ! [A,B] : ( v1_card_1(B) => ( ( r1_tarski(A,B) & r2_hidden(k1_card_1(A),k1_card_5(B)) ) => ( r2_hidden(k7_ordinal2(A),B) & r2_hidden(k3_tarski(A),B) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_card_5,t16_card_5,t17_wellord2,t18_card_5,t22_ordinal1,t26_ordinal2,t30_ordinal2,t21_zfrefle1,d2_card_5,t14_card_1,d8_xboole_0,t21_ordinal1,t23_ordinal1,t35_ordinal1,d3_tarski,d4_tarski,t23_ordinal1,t27_ordinal2,d2_ordinal1,t22_ordinal1]), [file(card_5,t38_card_5),interesting(0.85)]). fof(t46_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => ( ( v2_card_1(A) & r2_hidden(B,k1_card_5(A)) ) => k3_card_2(A,B) = k6_card_3(k2_card_5(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[s2_ordinal2,d3_tarski,d2_funct_2,d7_ordinal1,d8_ordinal2,t39_card_5,t20_zfrefle1,d5_funct_1,d2_funct_2,d4_tarski,t27_card_1,t54_card_3,t1_xboole_1,d3_card_2,t45_card_5,d2_card_3,t19_funcop_1,t23_ordinal1,d5_card_1,t24_card_1,d2_card_3,t67_funct_5,d3_card_2,t22_ordinal1,d3_card_5,d3_card_5,d5_funct_1,t92_zfmisc_1,t27_card_1,t55_card_3,d5_card_1,t1_xboole_1,t13_funcop_1,t43_card_3,t52_card_3,t1_xboole_1,d3_tarski,t23_ordinal1,t24_card_1,t22_ordinal1,d7_card_1,t22_card_4,t13_card_1,d3_card_5,d3_card_5,d5_funct_1,t92_zfmisc_1,t27_card_1,d5_card_1,t55_card_3,t31_card_5,t1_xboole_1,t1_xboole_1,d5_card_1,t36_card_1,t20_card_4,t27_card_5,d10_xboole_0]), [file(card_5,t46_card_5),interesting(0.85)]). fof(t18_card_5,theorem,( ! [A] : ( v3_ordinal1(A) => ? [B] : ( v1_card_1(B) & r1_tarski(B,k1_card_1(A)) & r4_zfrefle1(A,B) & ! [C] : ( v3_ordinal1(C) => ( r4_zfrefle1(A,C) => r1_ordinal1(B,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_card_5,s1_ordinal1,t17_card_5,t24_card_1,t1_xboole_1,t1_xboole_1,t21_zfrefle1,d10_xboole_0,d10_xboole_0,d10_xboole_0,t1_xboole_1]), [file(card_5,t18_card_5),interesting(0.84)]). fof(t15_card_5,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r1_tarski(B,A) => r4_zfrefle1(k7_ordinal2(B),k2_wellord2(k1_wellord2(B))) ) ) ), inference(mizar_proof,[status(thm)],[t14_card_5,d5_zfrefle1,t8_ordinal3,d7_ordinal2]), [file(card_5,t15_card_5),interesting(0.83)]). fof(t42_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ( v1_card_5(k3_card_1(0)) & v1_card_5(k2_card_1(A)) ) ) ), inference(mizar_proof,[status(thm)],[t34_card_5,d4_card_5,t35_card_5,d4_card_5]), [file(card_5,t42_card_5),interesting(0.82)]). fof(t11_card_5,theorem,( ! [A] : ( v3_ordinal1(A) => ~ v1_finset_1(k3_card_1(A)) ) ), inference(mizar_proof,[status(thm)],[t2_xboole_1,t43_card_1,t11_card_4]), [file(card_5,t11_card_5),interesting(0.81)]). fof(t24_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ? [B] : ( v3_ordinal1(B) & A = k3_card_1(B) ) ) ), inference(mizar_proof,[status(thm)],[t12_card_5]), [file(card_5,t24_card_5),interesting(0.80)]). fof(t45_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => r1_tarski(k6_card_3(k2_card_5(B,A)),k3_card_2(B,A)) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d3_card_5,t71_card_4,t13_funcop_1,t19_funcop_1,d3_tarski,d3_card_5,d10_xboole_0,d3_tarski,d3_card_5,l1_card_5,t25_card_5,t40_card_2,t52_card_3,t71_card_4,t5_funct_4,t27_card_5,t43_card_3,t46_card_3,t1_xboole_1]), [file(card_5,t45_card_5),interesting(0.79)]). fof(t31_card_5,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => r1_tarski(A,k3_card_2(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[l1_card_5,t25_card_5,t40_card_2,t71_card_4,t2_xboole_1]), [file(card_5,t31_card_5),interesting(0.79)]). fof(t9_card_5,theorem,( ! [A] : k2_card_1(k1_card_1(A)) = k2_card_1(A) ), inference(mizar_proof,[status(thm)],[d5_card_1,d6_card_1,d6_card_1]), [file(card_5,t9_card_5),interesting(0.78)]). fof(t23_card_5,theorem,( ! [A] : ( v1_card_1(A) => r2_hidden(A,k3_card_2(2,A)) ) ), inference(mizar_proof,[status(thm)],[t30_card_1,d5_card_1,t44_card_2]), [file(card_5,t23_card_5),interesting(0.77)]). fof(t34_card_5,theorem,( k1_card_5(k3_card_1(0)) = k3_card_1(0) ), inference(mizar_proof,[status(thm)],[d2_card_5,t13_card_1,t9_card_4,d5_card_1,d2_card_5,d5_zfrefle1,t83_card_1,t26_ordinal2,t26_finset_1,d10_xboole_0,t1_xboole_1,s2_card_3,t23_ordinal1,t34_ordinal1,t41_ordinal1,t19_ordinal2,t28_ordinal2,t7_ordinal1]), [file(card_5,t34_card_5),interesting(0.76)]). fof(t43_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => ( r1_tarski(A,B) => k3_card_2(A,B) = k3_card_2(2,B) ) ) ) ), inference(mizar_proof,[status(thm)],[l1_card_5,t25_card_5,t70_card_4,t30_card_1,d5_card_1,t44_card_2,t27_card_5,t43_card_2,t70_card_4,d10_xboole_0]), [file(card_5,t43_card_5),interesting(0.75)]). fof(t32_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => k3_tarski(A) = A ) ), inference(mizar_proof,[status(thm)],[t32_card_4,d6_ordinal1]), [file(card_5,t32_card_5),interesting(0.75)]). fof(t20_card_5,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ( v2_ordinal2(A) => v2_funct_1(A) ) ) ), inference(mizar_proof,[status(thm)],[d16_ordinal2,d8_funct_1,t23_ordinal1,t24_ordinal1]), [file(card_5,t20_card_5),interesting(0.73)]). fof(t16_card_5,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r1_tarski(B,A) => k1_card_1(B) = k1_card_1(k2_wellord2(k1_wellord2(B))) ) ) ), inference(mizar_proof,[status(thm)],[t9_wellord2,t17_wellord2,d2_wellord2,t50_wellord1,d9_wellord1,d1_wellord2,d7_wellord1,d4_wellord2,t21_card_1]), [file(card_5,t16_card_5),interesting(0.72)]). fof(t26_card_5,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => ( ( r1_tarski(B,A) | r2_hidden(B,A) ) => ( ~ v1_finset_1(A) & v1_card_1(A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_card_5,t13_card_1,t1_xboole_1,t11_card_4]), [file(card_5,t26_card_5),interesting(0.71)]). fof(t10_card_5,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r2_hidden(A,k3_card_3(B)) <=> ? [C] : ( r2_hidden(C,k1_relat_1(B)) & r2_hidden(A,k1_funct_1(B,C)) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_tarski,d5_funct_1,d5_funct_1,d4_tarski]), [file(card_5,t10_card_5),interesting(0.64)]). fof(t13_card_5,theorem,( ! [A] : ( v1_card_1(A) => ~ ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => A != k1_card_1(B) ) & ! [B] : ( v3_ordinal1(B) => A != k3_card_1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_card_1,t12_card_5,t4_card_4]), [file(card_5,t13_card_5),interesting(0.62)]). fof(t39_card_5,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) ) => ( ( k1_relat_1(C) = A & r1_tarski(k2_relat_1(C),B) & r2_hidden(A,k1_card_5(B)) ) => ( r2_hidden(k8_ordinal2(C),B) & r2_hidden(k3_card_3(C),B) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t28_card_1,d5_card_1,t22_ordinal1,t38_card_5]), [file(card_5,t39_card_5),interesting(0.62)]). fof(t36_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => r1_tarski(k3_card_1(0),k1_card_5(A)) ) ), inference(mizar_proof,[status(thm)],[d2_card_5,t32_card_4,t27_zfrefle1,t28_zfrefle1,t10_ordinal3,t83_card_1,d5_ordinal2]), [file(card_5,t36_card_5),interesting(0.61)]). fof(t19_card_5,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_ordinal2(B) ) => ( ( k2_relat_1(A) = k2_relat_1(B) & v2_ordinal2(A) & v2_ordinal2(B) ) => A = B ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t23_ordinal1,l20_card_5,d10_xboole_0,d3_tarski,t23_ordinal1,l20_card_5,t23_ordinal1,l20_card_5,t9_funct_1]), [file(card_5,t19_card_5),interesting(0.59)]). fof(t21_card_5,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_ordinal2(B) ) => k2_ordinal1(k1_ordinal4(A,B),k1_relat_1(A)) = A ) ) ), inference(mizar_proof,[status(thm)],[d1_ordinal4,t27_ordinal3,t28_xboole_1,t23_ordinal1,d1_ordinal4,t68_funct_1]), [file(card_5,t21_card_5),interesting(0.58)]). fof(t27_card_5,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => ( ( r1_tarski(B,A) | r2_hidden(B,A) ) => ( k1_card_2(B,A) = A & k1_card_2(A,B) = A & k2_card_2(B,A) = A & k2_card_2(A,B) = A ) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_card_5,t26_card_5,d5_card_1,t34_card_4,t78_card_4]), [file(card_5,t27_card_5),interesting(0.58)]). fof(l1_card_5,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_5,l1_card_5),interesting(0.57)]). fof(t41_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ~ ( r2_hidden(k1_card_5(A),A) & ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_ordinal2(B) ) => ~ ( k1_relat_1(B) = k1_card_5(A) & r1_tarski(k2_relat_1(B),A) & v2_ordinal2(B) & A = k8_ordinal2(B) & v1_relat_1(B) & v1_funct_1(B) & v1_card_3(B) & ~ r2_hidden(0,k2_relat_1(B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t40_card_5,d2_card_5,d5_zfrefle1,s4_ordinal1,t26_ordinal2,d8_ordinal2,d3_tarski,d5_funct_1,t23_ordinal1,d2_ordinal1,t91_relat_1,t27_ordinal2,d2_ordinal1,t91_relat_1,d5_funct_1,t25_card_1,t9_card_5,d7_card_1,t22_card_4,t146_relat_1,t25_card_1,t3_card_2,t22_ordinal1,d3_tarski,d5_funct_1,t23_ordinal1,t70_funct_1,t19_ordinal1,t38_card_5,t25_card_1,t9_card_5,d7_card_1,t22_card_4,t13_card_1,t15_ordinal3,s2_ordinal1,d3_tarski,d5_funct_1,t23_ordinal1,d16_ordinal2,t19_ordinal1,t73_funct_1,t36_card_1,t27_ordinal2,t19_ordinal1,d2_xboole_0,d3_tarski,t29_ordinal2,d5_funct_1,t23_ordinal1,d2_ordinal1,t91_relat_1,t36_card_1,d2_xboole_0,d2_ordinal1,d5_funct_1,t1_xboole_1,t27_ordinal2,t22_ordinal1,d10_xboole_0,d3_tarski,d5_funct_1,t23_ordinal1,t30_ordinal2,d1_card_3,t23_ordinal1,d2_ordinal1,t91_relat_1,t15_ordinal3,d5_funct_1,t23_ordinal1,d2_ordinal1,t91_relat_1,t15_ordinal3,t33_card_1]), [file(card_5,t41_card_5),interesting(0.57)]). fof(t28_card_5,theorem,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ( k1_card_2(A,A) = A & k2_card_2(A,A) = A ) ) ), inference(mizar_proof,[status(thm)],[t33_card_4,t77_card_4]), [file(card_5,t28_card_5),interesting(0.56)]). fof(t1_card_5,theorem, ( 1 = k1_tarski(0) & 2 = k2_tarski(0,1) ), inference(mizar_proof,[status(thm)],[d1_ordinal1,d1_ordinal1,t41_enumset1]), [file(card_5,t1_card_5),interesting(0.46)]). fof(t14_card_5,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ~ ( r1_tarski(B,A) & ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) ) => ~ ( C = k3_wellord1(k1_wellord2(k2_wellord2(k1_wellord2(B))),k1_wellord2(B)) & v2_ordinal2(C) & k1_relat_1(C) = k2_wellord2(k1_wellord2(B)) & k2_relat_1(C) = B ) ) ) ) ), inference(mizar_proof,[status(thm)],[t9_wellord2,t17_wellord2,d2_wellord2,t50_wellord1,d9_wellord1,d1_wellord2,d7_wellord1,d7_ordinal1,d8_ordinal2,d16_ordinal2,t19_ordinal1,d2_ordinal1,d1_wellord2,d5_funct_1,d8_funct_1,d1_wellord2,d8_xboole_0,t21_ordinal1,d7_wellord1]), [file(card_5,t14_card_5),interesting(0.37)]). fof(l20_card_5,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_ordinal2(B) ) => ( ( k2_relat_1(A) = k2_relat_1(B) & v2_ordinal2(A) & v2_ordinal2(B) ) => ! [C] : ( v3_ordinal1(C) => ( r2_hidden(C,k1_relat_1(A)) => ( r2_hidden(C,k1_relat_1(B)) & k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d16_ordinal2,d5_funct_1,d5_funct_1,t23_ordinal1,t19_ordinal1,d16_ordinal2,t19_ordinal1,d5_funct_1,d5_funct_1,t23_ordinal1,t7_ordinal1,t9_ordinal4,t26_ordinal1,t24_ordinal1,s2_ordinal1]), [file(card_5,l20_card_5),interesting(0.34)]). fof(t37_card_5,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( k1_card_5(0) = 0 & k1_card_5(k1_card_1(k1_nat_1(A,1))) = 1 ) ) ), inference(mizar_proof,[status(thm)],[d2_card_5,t2_xboole_1,d10_xboole_0,d2_card_5,t52_card_1,d5_card_1,t20_card_2,t25_zfrefle1,d2_card_5,t10_ordinal1,t19_ordinal3,t2_xboole_1,t28_zfrefle1]), [file(card_5,t37_card_5),interesting(0.32)]). fof(t25_card_5,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_finset_1(B) & v1_card_1(B) ) => ( B != 0 & B != 1 & B != 2 & B != k1_card_1(A) & r2_hidden(k1_card_1(A),B) & r1_tarski(k3_card_1(0),B) ) ) ) ), inference(mizar_proof,[status(thm)],[t9_card_4,t11_card_4]), [file(card_5,t25_card_5),interesting(0.30)]). 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(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(t39_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => k3_card_1(k1_ordinal1(A)) = k2_card_1(k3_card_1(A)) ) ), file(card_1,t39_card_1), [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(t42_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( k3_card_1(A) = k3_card_1(B) => A = B ) ) ) ), file(card_1,t42_card_1), [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(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(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(t23_ordinal1,theorem,( ! [A,B] : ( v3_ordinal1(B) => ( r2_hidden(A,B) => v3_ordinal1(A) ) ) ), file(ordinal1,t23_ordinal1), [interesting(0.00)]). fof(t41_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r2_hidden(A,B) <=> r2_hidden(k3_card_1(A),k3_card_1(B)) ) ) ) ), file(card_1,t41_card_1), [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(t43_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r1_ordinal1(A,B) <=> r1_tarski(k3_card_1(A),k3_card_1(B)) ) ) ) ), file(card_1,t43_card_1), [interesting(0.00)]). fof(t11_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) <=> r1_ordinal1(k3_card_1(0),A) ) ) ), file(card_4,t11_card_4), [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(t31_ordinal1,theorem,( ! [A] : ( ! [B] : ( r2_hidden(B,A) => ( v3_ordinal1(B) & r1_tarski(B,A) ) ) => v3_ordinal1(A) ) ), file(ordinal1,t31_ordinal1), [interesting(0.00)]). fof(s2_ordinal2,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & k1_relat_1(A) = f1_s2_ordinal2 & ! [B] : ( v3_ordinal1(B) => ( r2_hidden(B,f1_s2_ordinal2) => k1_funct_1(A,B) = f2_s2_ordinal2(B) ) ) ) ), file(ordinal2,s2_ordinal2), [interesting(0.00)]). fof(d7_card_1,definition,( ! [A] : ( v1_card_1(A) => ( v2_card_1(A) <=> ! [B] : ( v1_card_1(B) => A != k2_card_1(B) ) ) ) ), file(card_1,d7_card_1), [interesting(0.00)]). 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) ) ) ) ), file(card_4,t22_card_4), [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(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(t40_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => ( v4_ordinal1(A) => ( A = k1_xboole_0 | ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) ) => ( ( k1_relat_1(B) = A & ! [C] : ( v3_ordinal1(C) => ( r2_hidden(C,A) => k1_funct_1(B,C) = k3_card_1(C) ) ) ) => k3_card_1(A) = k1_card_1(k8_ordinal2(B)) ) ) ) ) ) ), file(card_1,t40_card_1), [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(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(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(t65_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ( k1_relat_1(A) = k1_xboole_0 <=> k2_relat_1(A) = k1_xboole_0 ) ) ), file(relat_1,t65_relat_1), [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(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(s1_card_1,theorem, ( ( p1_s1_card_1(k1_xboole_0) & ! [A] : ( v1_card_1(A) => ( p1_s1_card_1(A) => p1_s1_card_1(k2_card_1(A)) ) ) & ! [A] : ( v1_card_1(A) => ( ( v2_card_1(A) & ! [B] : ( v1_card_1(B) => ( r2_hidden(B,A) => p1_s1_card_1(B) ) ) ) => ( A = k1_xboole_0 | p1_s1_card_1(A) ) ) ) ) => ! [A] : ( v1_card_1(A) => p1_s1_card_1(A) ) ), file(card_1,s1_card_1), [interesting(0.00)]). 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) ) ) ), file(card_4,t4_card_4), [interesting(0.00)]). fof(d1_ordinal1,definition,( ! [A] : k1_ordinal1(A) = k2_xboole_0(A,k1_tarski(A)) ), file(ordinal1,d1_ordinal1), [interesting(0.00)]). fof(t41_enumset1,theorem,( ! [A,B] : k2_tarski(A,B) = k2_xboole_0(k1_tarski(A),k1_tarski(B)) ), file(enumset1,t41_enumset1), [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(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(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(d16_ordinal2,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ( v2_ordinal2(A) <=> ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( r2_hidden(B,C) & r2_hidden(C,k1_relat_1(A)) ) => r2_hidden(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) ) ), file(ordinal2,d16_ordinal2), [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(t9_ordinal4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( v2_ordinal2(A) & r1_ordinal1(B,C) & r2_hidden(C,k1_relat_1(A)) ) => r1_ordinal1(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) ), file(ordinal4,t9_ordinal4), [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(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(s2_ordinal1,theorem, ( ! [A] : ( v3_ordinal1(A) => ( ! [B] : ( v3_ordinal1(B) => ( r2_hidden(B,A) => p1_s2_ordinal1(B) ) ) => p1_s2_ordinal1(A) ) ) => ! [A] : ( v3_ordinal1(A) => p1_s2_ordinal1(A) ) ), file(ordinal1,s2_ordinal1), [interesting(0.00)]). fof(t9_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) => A = B ) ) ) ), file(funct_1,t9_funct_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(t3_card_2,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(k1_card_1(k9_relat_1(B,A)),k1_card_1(A)) ) ), file(card_2,t3_card_2), [interesting(0.00)]). fof(t60_card_3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(k1_card_1(A),B) => r2_hidden(A,B) ) ) ) ), file(card_3,t60_card_3), [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(t97_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( r1_tarski(k1_relat_1(B),A) => k7_relat_1(B,A) = B ) ) ), file(relat_1,t97_relat_1), [interesting(0.00)]). fof(t72_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,A) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t72_funct_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(t9_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r1_tarski(B,A) => v2_wellord1(k1_wellord2(B)) ) ) ), file(wellord2,t9_wellord2), [interesting(0.00)]). fof(t17_wellord2,theorem,( ! [A,B] : ( v3_ordinal1(B) => ( r1_tarski(A,B) => r1_ordinal1(k2_wellord2(k1_wellord2(A)),B) ) ) ), file(wellord2,t17_wellord2), [interesting(0.00)]). fof(d2_wellord2,definition,( ! [A] : ( v1_relat_1(A) => ( v2_wellord1(A) => ! [B] : ( v3_ordinal1(B) => ( B = k2_wellord2(A) <=> r4_wellord1(A,k1_wellord2(B)) ) ) ) ) ), file(wellord2,d2_wellord2), [interesting(0.00)]). fof(t50_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r4_wellord1(A,B) => r4_wellord1(B,A) ) ) ) ), file(wellord1,t50_wellord1), [interesting(0.00)]). fof(d9_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( ( v2_wellord1(A) & r4_wellord1(A,B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( C = k3_wellord1(A,B) <=> r3_wellord1(A,B,C) ) ) ) ) ) ), file(wellord1,d9_wellord1), [interesting(0.00)]). fof(d1_wellord2,definition,( ! [A,B] : ( v1_relat_1(B) => ( B = k1_wellord2(A) <=> ( k3_relat_1(B) = A & ! [C,D] : ( ( r2_hidden(C,A) & r2_hidden(D,A) ) => ( r2_hidden(k4_tarski(C,D),B) <=> r1_tarski(C,D) ) ) ) ) ) ), file(wellord2,d1_wellord2), [interesting(0.00)]). fof(d7_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r3_wellord1(A,B,C) <=> ( k1_relat_1(C) = k3_relat_1(A) & k2_relat_1(C) = k3_relat_1(B) & v2_funct_1(C) & ! [D,E] : ( r2_hidden(k4_tarski(D,E),A) <=> ( r2_hidden(D,k3_relat_1(A)) & r2_hidden(E,k3_relat_1(A)) & r2_hidden(k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E)),B) ) ) ) ) ) ) ) ), file(wellord1,d7_wellord1), [interesting(0.00)]). fof(d7_ordinal1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v5_ordinal1(A) <=> v3_ordinal1(k1_relat_1(A)) ) ) ), file(ordinal1,d7_ordinal1), [interesting(0.00)]). fof(d8_ordinal2,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_ordinal2(A) <=> ? [B] : ( v3_ordinal1(B) & r1_tarski(k2_relat_1(A),B) ) ) ) ), file(ordinal2,d8_ordinal2), [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(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(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(d1_ordinal4,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) ) => ( C = k1_ordinal4(A,B) <=> ( k1_relat_1(C) = k14_ordinal2(k1_relat_1(A),k1_relat_1(B)) & ! [D] : ( v3_ordinal1(D) => ( r2_hidden(D,k1_relat_1(A)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) & ! [D] : ( v3_ordinal1(D) => ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(C,k14_ordinal2(k1_relat_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ), file(ordinal4,d1_ordinal4), [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(t28_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), [interesting(0.00)]). fof(t68_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( B = k7_relat_1(C,A) <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A) & ! [D] : ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ), file(funct_1,t68_funct_1), [interesting(0.00)]). fof(d6_ordinal3,definition,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( r1_ordinal1(B,A) => ( C = k5_ordinal3(A,B) <=> A = k14_ordinal2(B,C) ) ) & ( ~ r1_ordinal1(B,A) => ( C = k5_ordinal3(A,B) <=> C = k1_xboole_0 ) ) ) ) ) ) ), file(ordinal3,d6_ordinal3), [interesting(0.00)]). fof(t66_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( r2_hidden(A,B) => ( ( ~ r1_ordinal1(C,A) & ~ r2_hidden(C,A) ) | r2_hidden(k5_ordinal3(A,C),k5_ordinal3(B,C)) ) ) ) ) ) ), file(ordinal3,t66_ordinal3), [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(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(t73_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( r2_hidden(B,k1_relat_1(C)) & r2_hidden(B,A) ) => r2_hidden(k1_funct_1(C,B),k2_relat_1(k7_relat_1(C,A))) ) ) ), file(funct_1,t73_funct_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(d2_funct_2,definition,( ! [A,B,C] : ( C = k1_funct_2(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( v1_relat_1(E) & v1_funct_1(E) & D = E & k1_relat_1(E) = A & r1_tarski(k2_relat_1(E),B) ) ) ) ), file(funct_2,d2_funct_2), [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(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(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(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(t33_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) => k1_card_2(A,A) = A ) ) ), file(card_4,t33_card_4), [interesting(0.00)]). fof(t77_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) => k2_card_2(A,A) = A ) ) ), file(card_4,t77_card_4), [interesting(0.00)]). fof(t29_card_5,theorem,( $true ), file(card_5,t29_card_5), [interesting(0.00)]). fof(t2_card_5,theorem,( $true ), file(card_5,t2_card_5), [interesting(0.00)]). fof(t30_card_5,theorem,( $true ), file(card_5,t30_card_5), [interesting(0.00)]). fof(t33_card_5,theorem,( $true ), file(card_5,t33_card_5), [interesting(0.00)]). fof(d2_card_5,definition,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( B = k1_card_5(A) <=> ( r4_zfrefle1(A,B) & ! [C] : ( v1_card_1(C) => ( r4_zfrefle1(A,C) => r1_tarski(B,C) ) ) ) ) ) ) ), file(card_5,d2_card_5), [interesting(0.00)]). fof(t32_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( ~ v1_finset_1(A) => v4_ordinal1(A) ) ) ), file(card_4,t32_card_4), [interesting(0.00)]). fof(t27_zfrefle1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r4_zfrefle1(A,B) => ( v4_ordinal1(A) <=> v4_ordinal1(B) ) ) ) ) ), file(zfrefle1,t27_zfrefle1), [interesting(0.00)]). fof(t28_zfrefle1,theorem,( ! [A] : ( v3_ordinal1(A) => ( r4_zfrefle1(A,k1_xboole_0) => A = k1_xboole_0 ) ) ), file(zfrefle1,t28_zfrefle1), [interesting(0.00)]). fof(t83_card_1,theorem,( k3_card_1(0) = k5_ordinal2 ), file(card_1,t83_card_1), [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(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(t20_card_2,theorem,( 1 = k4_ordinal2 ), file(card_2,t20_card_2), [interesting(0.00)]). fof(t25_zfrefle1,theorem,( ! [A] : ( v3_ordinal1(A) => r4_zfrefle1(k1_ordinal1(A),k4_ordinal2) ) ), file(zfrefle1,t25_zfrefle1), [interesting(0.00)]). fof(t10_ordinal1,theorem,( ! [A] : r2_hidden(A,k1_ordinal1(A)) ), file(ordinal1,t10_ordinal1), [interesting(0.00)]). fof(t19_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ~ ( r1_ordinal1(A,k4_ordinal2) & A != k1_xboole_0 & A != k4_ordinal2 ) ) ), file(ordinal3,t19_ordinal3), [interesting(0.00)]). fof(t3_card_5,theorem,( $true ), file(card_5,t3_card_5), [interesting(0.00)]). fof(t9_card_4,theorem,( ! [A] : ( v1_card_1(A) => ( v1_finset_1(A) <=> r2_hidden(A,k3_card_1(0)) ) ) ), file(card_4,t9_card_4), [interesting(0.00)]). fof(d5_zfrefle1,definition,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r4_zfrefle1(A,B) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) & k1_relat_1(C) = B & r1_tarski(k2_relat_1(C),A) & v2_ordinal2(C) & A = k8_ordinal2(C) ) ) ) ) ), file(zfrefle1,d5_zfrefle1), [interesting(0.00)]). fof(t26_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => k7_ordinal2(A) = A ) ), file(ordinal2,t26_ordinal2), [interesting(0.00)]). fof(t26_finset_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_finset_1(k1_relat_1(A)) => v1_finset_1(k2_relat_1(A)) ) ) ), file(finset_1,t26_finset_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(s2_card_3,theorem, ( ( f1_s2_card_3 != k1_xboole_0 & ! [A,B] : ( ( p1_s2_card_3(A,B) & p1_s2_card_3(B,A) ) => A = B ) & ! [A,B,C] : ( ( p1_s2_card_3(A,B) & p1_s2_card_3(B,C) ) => p1_s2_card_3(A,C) ) ) => ? [A] : ( r2_hidden(A,f1_s2_card_3) & ! [B] : ~ ( r2_hidden(B,f1_s2_card_3) & B != A & p1_s2_card_3(B,A) ) ) ), file(card_3,s2_card_3), [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(t28_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ! [C] : ( v3_ordinal1(C) => ( r2_hidden(C,B) => r2_hidden(C,A) ) ) => r1_ordinal1(k7_ordinal2(B),A) ) ) ), file(ordinal2,t28_ordinal2), [interesting(0.00)]). fof(d4_card_5,definition,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ( v1_card_5(A) <=> k1_card_5(A) = A ) ) ), file(card_5,d4_card_5), [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(t24_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => r1_ordinal1(k1_card_1(A),A) ) ), file(card_1,t24_card_1), [interesting(0.00)]). 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) ) ) ) ), file(card_4,t23_card_4), [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(t52_card_3,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => k6_card_3(k2_funcop_1(A,B)) = k2_card_2(A,B) ) ) ), file(card_3,t52_card_3), [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(t8_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r1_tarski(B,A) => k2_ordinal2(B) = B ) ) ), file(ordinal3,t8_ordinal3), [interesting(0.00)]). fof(t27_zf_refle,theorem,( ! [A] : r1_tarski(k7_ordinal2(A),k1_ordinal1(k3_tarski(k2_ordinal2(A)))) ), file(zf_refle,t27_zf_refle), [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(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)) ) ) ) ) ) ), file(card_4,t42_card_4), [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(t19_card_2,theorem, ( 0 = k1_card_1(k1_xboole_0) & 1 = k1_card_1(k4_ordinal2) & 2 = k1_card_1(k1_ordinal1(k4_ordinal2)) ), file(card_2,t19_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(t29_card_2,theorem,( ! [A] : ( v1_card_1(A) => k1_card_2(A,0) = A ) ), file(card_2,t29_card_2), [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(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(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 ) ) ) ) ), file(card_4,t34_card_4), [interesting(0.00)]). 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 ) ) ) ) ) ), file(card_4,t78_card_4), [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(t30_card_1,theorem,( ! [A] : r2_hidden(k1_card_1(A),k1_card_1(k1_zfmisc_1(A))) ), file(card_1,t30_card_1), [interesting(0.00)]). fof(t44_card_2,theorem,( ! [A] : k3_card_2(2,k1_card_1(A)) = k1_card_1(k1_zfmisc_1(A)) ), file(card_2,t44_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)) ) ) ) ) ) ), file(card_4,t70_card_4), [interesting(0.00)]). fof(t43_card_2,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( v1_card_1(C) => k3_card_2(A,k2_card_2(B,C)) = k3_card_2(k3_card_2(A,B),C) ) ) ) ), file(card_2,t43_card_2), [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(d7_ordinal2,definition,( ! [A,B] : ( v3_ordinal1(B) => ( B = k7_ordinal2(A) <=> ( r1_tarski(k2_ordinal2(A),B) & ! [C] : ( v3_ordinal1(C) => ( r1_tarski(k2_ordinal2(A),C) => r1_ordinal1(B,C) ) ) ) ) ) ), file(ordinal2,d7_ordinal2), [interesting(0.00)]). fof(t45_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ), file(relat_1,t45_relat_1), [interesting(0.00)]). fof(t46_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(k2_relat_1(A),k1_relat_1(B)) => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ), file(relat_1,t46_relat_1), [interesting(0.00)]). fof(t46_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( v2_funct_1(A) & v2_funct_1(B) ) => v2_funct_1(k5_relat_1(A,B)) ) ) ) ), file(funct_1,t46_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(t22_funct_1,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(k5_relat_1(C,B))) => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ), file(funct_1,t22_funct_1), [interesting(0.00)]). fof(t27_ordinal2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r2_hidden(A,B) => r2_hidden(A,k7_ordinal2(B)) ) ) ), file(ordinal2,t27_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(t90_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ), file(relat_1,t90_relat_1), [interesting(0.00)]). fof(t70_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A))) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t70_funct_1), [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(s1_ordinal1,theorem, ( ? [A] : ( v3_ordinal1(A) & p1_s1_ordinal1(A) ) => ? [A] : ( v3_ordinal1(A) & p1_s1_ordinal1(A) & ! [B] : ( v3_ordinal1(B) => ( p1_s1_ordinal1(B) => r1_ordinal1(A,B) ) ) ) ), file(ordinal1,s1_ordinal1), [interesting(0.00)]). fof(t30_ordinal2,theorem,( ! [A,B] : ( r1_tarski(A,B) => r1_ordinal1(k7_ordinal2(A),k7_ordinal2(B)) ) ), file(ordinal2,t30_ordinal2), [interesting(0.00)]). fof(t21_zfrefle1,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( r4_zfrefle1(A,B) & r4_zfrefle1(B,C) ) => r4_zfrefle1(A,C) ) ) ) ) ), file(zfrefle1,t21_zfrefle1), [interesting(0.00)]). fof(t35_ordinal1,theorem,( ! [A] : ( ! [B] : ( r2_hidden(B,A) => v3_ordinal1(B) ) => v3_ordinal1(k3_tarski(A)) ) ), file(ordinal1,t35_ordinal1), [interesting(0.00)]). fof(t20_zfrefle1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => r1_tarski(k2_relat_1(A),k8_ordinal2(A)) ) ), file(zfrefle1,t20_zfrefle1), [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(t68_funct_5,theorem,( ! [A,B,C,D] : ( ( k1_card_1(A) = k1_card_1(B) & k1_card_1(C) = k1_card_1(D) ) => k1_card_1(k1_funct_2(A,C)) = k1_card_1(k1_funct_2(B,D)) ) ), file(funct_5,t68_funct_5), [interesting(0.00)]). fof(t40_card_2,theorem,( ! [A] : ( v1_card_1(A) => ( k3_card_2(A,1) = A & k3_card_2(1,A) = 1 ) ) ), file(card_2,t40_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)) ) ) ) ) ) ) ), file(card_4,t68_card_4), [interesting(0.00)]). fof(d3_card_5,definition,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_card_3(C) ) => ( C = k2_card_5(A,B) <=> ( ! [D] : ( r2_hidden(D,k1_relat_1(C)) <=> ( r2_hidden(D,A) & v1_card_1(D) ) ) & ! [D] : ( v1_card_1(D) => ( r2_hidden(D,A) => k1_funct_1(C,D) = k3_card_2(D,B) ) ) ) ) ) ) ) ), file(card_5,d3_card_5), [interesting(0.00)]). 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)) ) ) ) ) ) ), file(card_4,t71_card_4), [interesting(0.00)]). fof(t5_funct_4,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k2_funcop_1(A,C),k2_funcop_1(B,C)) ) ), file(funct_4,t5_funct_4), [interesting(0.00)]). fof(t46_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) ) => ( r1_tarski(A,B) => r1_tarski(k6_card_3(A),k6_card_3(B)) ) ) ) ), file(card_3,t46_card_3), [interesting(0.00)]). fof(t67_funct_5,theorem,( ! [A,B,C,D] : ( ( r2_wellord2(A,B) & r2_wellord2(C,D) ) => ( r2_wellord2(k1_funct_2(A,C),k1_funct_2(B,D)) & k1_card_1(k1_funct_2(A,C)) = k1_card_1(k1_funct_2(B,D)) ) ) ), file(funct_5,t67_funct_5), [interesting(0.00)]). fof(t55_card_3,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_card_3(A) ) => r1_tarski(k1_card_1(k3_card_3(A)),k6_card_3(A)) ) ), file(card_3,t55_card_3), [interesting(0.00)]). fof(t36_card_1,theorem,( ! [A] : ( v3_ordinal1(A) => r2_hidden(A,k2_card_1(A)) ) ), file(card_1,t36_card_1), [interesting(0.00)]). fof(t20_card_4,theorem,( ! [A,B] : ( k1_card_1(A) = k1_card_1(B) <=> k2_card_1(A) = k2_card_1(B) ) ), file(card_4,t20_card_4), [interesting(0.00)]). fof(s4_ordinal1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & k1_relat_1(A) = f1_s4_ordinal1 & ! [B] : ( v3_ordinal1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) ) => ( ( r2_hidden(B,f1_s4_ordinal1) & C = k2_ordinal1(A,B) ) => k1_funct_1(A,B) = f2_s4_ordinal1(C) ) ) ) ) ), file(ordinal1,s4_ordinal1), [interesting(0.00)]). fof(t25_card_1,theorem,( ! [A,B] : ( v1_card_1(B) => ( r2_hidden(A,B) => r2_hidden(k1_card_1(A),B) ) ) ), file(card_1,t25_card_1), [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(t146_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => k9_relat_1(A,k1_relat_1(A)) = k2_relat_1(A) ) ), file(relat_1,t146_relat_1), [interesting(0.00)]). fof(t15_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( k2_xboole_0(A,B) = A | k2_xboole_0(A,B) = B ) ) ) ), file(ordinal3,t15_ordinal3), [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(d1_card_3,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_card_3(A) <=> ! [B] : ( r2_hidden(B,k1_relat_1(A)) => v1_card_1(k1_funct_1(A,B)) ) ) ) ), file(card_3,d1_card_3), [interesting(0.00)]). fof(t33_card_1,theorem,( ! [A] : r2_hidden(k1_card_1(k1_xboole_0),k2_card_1(A)) ), file(card_1,t33_card_1), [interesting(0.00)]). fof(s1_partfun1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s1_partfun1 & ! [B] : ( r2_hidden(B,f1_s1_partfun1) => ( ( p1_s1_partfun1(B) => k1_funct_1(A,B) = f2_s1_partfun1(B) ) & ( ~ p1_s1_partfun1(B) => k1_funct_1(A,B) = f3_s1_partfun1(B) ) ) ) ) ), file(partfun1,s1_partfun1), [interesting(0.00)]). fof(s1_zfrefle1,theorem, ( ! [A] : ~ ( r2_hidden(A,f1_s1_zfrefle1) & ! [B] : ~ p1_s1_zfrefle1(A,B) ) => ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s1_zfrefle1 & ! [B] : ( r2_hidden(B,f1_s1_zfrefle1) => p1_s1_zfrefle1(B,k1_funct_1(A,B)) ) ) ), file(zfrefle1,s1_zfrefle1), [interesting(0.00)]). fof(d6_ordinal1,definition,( ! [A] : ( v4_ordinal1(A) <=> A = k3_tarski(A) ) ), file(ordinal1,d6_ordinal1), [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(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(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(t33_card_3,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r2_hidden(A,k3_card_3(k2_card_3(B))) <=> ( r2_hidden(k2_mcart_1(A),k1_relat_1(B)) & r2_hidden(k1_mcart_1(A),k1_funct_1(B,k2_mcart_1(A))) & A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ) ) ), file(card_3,t33_card_3), [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(d3_card_3,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( B = k2_card_3(A) <=> ( k1_relat_1(B) = k1_relat_1(A) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(B,C) = k2_zfmisc_1(k1_funct_1(A,C),k1_tarski(C)) ) ) ) ) ) ), file(card_3,d3_card_3), [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(t4_card_5,theorem,( $true ), file(card_5,t4_card_5), [interesting(0.00)]). fof(t5_card_5,theorem,( $true ), file(card_5,t5_card_5), [interesting(0.00)]). fof(t6_card_5,theorem,( $true ), file(card_5,t6_card_5), [interesting(0.00)]). fof(t7_card_5,theorem,( $true ), file(card_5,t7_card_5), [interesting(0.00)]). fof(t30_axioms,theorem,( ! [A] : ( v4_ordinal2(A) => A = a_1_0_axioms(A) ) ), file(axioms,t30_axioms), [interesting(0.00)]). fof(d1_finseq_1,definition,( ! [A] : ( v4_ordinal2(A) => k1_finseq_1(A) = a_1_0_finseq_1(A) ) ), file(finseq_1,d1_finseq_1), [interesting(0.00)]). fof(t38_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), [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(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(t8_card_5,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_finseq_1(A) = k4_xboole_0(k1_nat_1(A,1),k1_tarski(0)) ) ), inference(mizar_proof,[status(thm)],[t30_axioms,d1_finseq_1,d3_tarski,t38_nat_1,d1_tarski,d4_xboole_0,d10_xboole_0,d3_tarski,d4_xboole_0,d1_tarski,t38_nat_1]), [file(card_5,t8_card_5),interesting(0.00)]).