fof(t14_wellord2,theorem,( ! [A] : ( v1_relat_1(A) => ~ ( v2_wellord1(A) & ! [B] : ( v3_ordinal1(B) => ~ r4_wellord1(A,k1_wellord2(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[s1_xboole_0,t32_wellord1,t19_wellord1,t35_wellord1,d1_wellord1,t37_wellord1,t29_wellord1,t13_wellord2,t41_wellord1,t13_wellord2]), [file(wellord2,t14_wellord2),interesting(1.00)]). fof(t26_wellord2,theorem,( ! [A] : ? [B] : ( v1_relat_1(B) & r2_wellord1(B,A) ) ), inference(mizar_proof,[status(thm)],[t136_zfmisc_1,s1_xboole_0,d3_tarski,t23_ordinal1,d2_ordinal1,t31_ordinal1,d3_tarski,d1_wellord2,t7_wellord2,l30_wellord2,t25_wellord2,s3_funct_1,d3_tarski,d5_funct_1,d3_tarski,d1_tarski,d4_wellord2,d8_funct_1,t6_zfmisc_1,t32_wellord1,t39_wellord1,l30_wellord2]), [file(wellord2,t26_wellord2),interesting(0.89)]). fof(t7_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => v2_wellord1(k1_wellord2(A)) ) ), inference(mizar_proof,[status(thm)],[t2_wellord2,t3_wellord2,t4_wellord2,t5_wellord2,t6_wellord2,d4_wellord1]), [file(wellord2,t7_wellord2),interesting(0.79)]). fof(t3_wellord2,theorem,( ! [A] : v8_relat_2(k1_wellord2(A)) ), inference(mizar_proof,[status(thm)],[d8_relat_2,d16_relat_2,d1_wellord2,d1_wellord2,t1_xboole_1,d1_wellord2]), [file(wellord2,t3_wellord2),interesting(0.78)]). fof(t5_wellord2,theorem,( ! [A] : v4_relat_2(k1_wellord2(A)) ), inference(mizar_proof,[status(thm)],[d4_relat_2,d12_relat_2,d1_wellord2,d10_xboole_0]), [file(wellord2,t5_wellord2),interesting(0.77)]). fof(t9_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r1_tarski(B,A) => v2_wellord1(k1_wellord2(B)) ) ) ), inference(mizar_proof,[status(thm)],[t7_wellord2,t32_wellord1,t8_wellord2]), [file(wellord2,t9_wellord2),interesting(0.76)]). fof(t2_wellord2,theorem,( ! [A] : v1_relat_2(k1_wellord2(A)) ), inference(mizar_proof,[status(thm)],[d1_relat_2,d9_relat_2,d1_wellord2]), [file(wellord2,t2_wellord2),interesting(0.75)]). fof(t6_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => v1_wellord1(k1_wellord2(A)) ) ), inference(mizar_proof,[status(thm)],[d2_wellord1,d1_wellord2,d3_tarski,t23_ordinal1,s1_ordinal1,d7_xboole_0,d3_xboole_0,d1_wellord1,t23_ordinal1,d1_wellord2,d10_xboole_0]), [file(wellord2,t6_wellord2),interesting(0.74)]). fof(t4_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => v6_relat_2(k1_wellord2(A)) ) ), inference(mizar_proof,[status(thm)],[d6_relat_2,d14_relat_2,d1_wellord2,t23_ordinal1]), [file(wellord2,t4_wellord2),interesting(0.71)]). fof(t11_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r4_wellord1(k1_wellord2(A),k1_wellord2(B)) => A = B ) ) ) ), inference(mizar_proof,[status(thm)],[t24_ordinal1,t10_wellord2,d1_wellord2,t13_wellord1,t8_wellord2,t7_wellord2,t57_wellord1,t50_wellord1,t10_wellord2,d1_wellord2,t13_wellord1,t8_wellord2,t7_wellord2,t57_wellord1]), [file(wellord2,t11_wellord2),interesting(0.71)]). fof(t8_wellord2,theorem,( ! [A,B] : ( r1_tarski(A,B) => k2_wellord1(k1_wellord2(B),A) = k1_wellord2(A) ) ), inference(mizar_proof,[status(thm)],[d2_relat_1,t16_wellord1,t106_zfmisc_1,d1_wellord2,d1_wellord2,d1_wellord2,t30_relat_1,d1_wellord2,d1_wellord2,t106_zfmisc_1,t16_wellord1]), [file(wellord2,t8_wellord2),interesting(0.66)]). fof(t22_wellord2,theorem,( ! [A,B,C] : ( ( r2_wellord2(A,B) & r2_wellord2(B,C) ) => r2_wellord2(A,C) ) ), inference(mizar_proof,[status(thm)],[d4_wellord2,d4_wellord2,d4_wellord2,t46_funct_1,t46_relat_1,t47_relat_1]), [file(wellord2,t22_wellord2),interesting(0.64)]). fof(t12_wellord2,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( r4_wellord1(A,k1_wellord2(B)) & r4_wellord1(A,k1_wellord2(C)) ) => B = C ) ) ) ) ), inference(mizar_proof,[status(thm)],[t50_wellord1,t52_wellord1,t11_wellord2]), [file(wellord2,t12_wellord2),interesting(0.63)]). fof(t17_wellord2,theorem,( ! [A,B] : ( v3_ordinal1(B) => ( r1_tarski(A,B) => r1_ordinal1(k2_wellord2(k1_wellord2(A)),B) ) ) ), inference(mizar_proof,[status(thm)],[t9_wellord2,t7_wellord2,d1_wellord2,t8_wellord2,t50_wellord1,d2_wellord2,t23_ordinal1,t10_wellord2,d2_ordinal1,t8_wellord2,t50_wellord1,d2_wellord2,t64_wellord1]), [file(wellord2,t17_wellord2),interesting(0.63)]). fof(l30_wellord2,theorem,( ! [A,B] : ( v1_relat_1(B) => ~ ( v2_wellord1(B) & r2_wellord2(A,k3_relat_1(B)) & ! [C] : ( v1_relat_1(C) => ~ r2_wellord1(C,A) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_wellord1,d9_relat_2,d4_wellord2,s1_relat_1,d3_tarski,d4_relat_1,d5_relat_1,d2_xboole_0,d10_xboole_0,d3_tarski,d5_funct_1,d1_relat_2,t30_relat_1,d7_wellord1,t49_wellord1,t54_wellord1,t8_wellord1]), [file(wellord2,l30_wellord2),interesting(0.62)]). fof(t10_wellord2,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r2_hidden(A,B) => A = k1_wellord1(k1_wellord2(B),A) ) ) ) ), inference(mizar_proof,[status(thm)],[t23_ordinal1,d2_ordinal1,d2_ordinal1,d1_wellord2,d1_wellord1,d3_tarski,d10_xboole_0,d3_tarski,d1_wellord1,d1_wellord2,t30_relat_1,t23_ordinal1,d1_wellord2,d8_xboole_0,t21_ordinal1]), [file(wellord2,t10_wellord2),interesting(0.61)]). fof(t25_wellord2,theorem,( ! [A,B] : ( v1_relat_1(B) => ( r2_wellord1(B,A) => ( k3_relat_1(k2_wellord1(B,A)) = A & v2_wellord1(k2_wellord1(B,A)) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_wellord1,t20_wellord1,d10_xboole_0,d3_tarski,d1_relat_2,t106_zfmisc_1,t16_wellord1,t30_relat_1,d1_relat_2,d1_relat_2,t106_zfmisc_1,t16_wellord1,d5_wellord1,d8_relat_2,t16_wellord1,d8_relat_2,t106_zfmisc_1,t16_wellord1,d4_relat_2,t16_wellord1,d4_relat_2,d6_relat_2,d6_relat_2,t106_zfmisc_1,t16_wellord1,d3_wellord1,d3_wellord1,t3_xboole_0,d1_wellord1,t16_wellord1,d1_wellord1,t3_xboole_0,t8_wellord1]), [file(wellord2,t25_wellord2),interesting(0.59)]). fof(t13_wellord2,theorem,( ! [A] : ( v1_relat_1(A) => ~ ( v2_wellord1(A) & ! [B] : ~ ( r2_hidden(B,k3_relat_1(A)) & ! [C] : ( v3_ordinal1(C) => ~ r4_wellord1(k2_wellord1(A,k1_wellord1(A,B)),k1_wellord2(C)) ) ) & ! [B] : ( v3_ordinal1(B) => ~ r4_wellord1(A,k1_wellord2(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t12_wellord2,s2_funct_1,d5_funct_1,t36_ordinal1,d1_wellord2,t7_wellord2,d5_funct_1,d1_wellord2,t30_relat_1,t23_ordinal1,d1_wellord2,d8_xboole_0,d1_wellord2,t21_ordinal1,t7_wellord2,t50_wellord1,d9_wellord1,t61_wellord1,t19_wellord1,d1_wellord1,t10_wellord2,t8_wellord2,t35_wellord1,t37_wellord1,t29_wellord1,t50_wellord1,d5_funct_1,t23_ordinal1,t10_wellord2,t36_wellord1,d5_funct_1,t23_ordinal1,d8_wellord1,d7_wellord1,d1_wellord2,d8_funct_1,t50_wellord1,t52_wellord1,t58_wellord1,t30_relat_1,d1_wellord2,d5_funct_1,t32_wellord1,t13_wellord1,d1_wellord1,t39_wellord1,t7_wellord2,t10_wellord2,d2_ordinal1,t8_wellord2,t8_wellord2,t35_wellord1,t37_wellord1,t29_wellord1,t62_wellord1,d1_wellord2,d4_wellord1,d14_relat_2,d4_wellord1,d9_relat_2,t5_wellord2,d12_relat_2,d1_relat_2,d6_relat_2,t32_wellord1,t13_wellord1,d1_wellord1,t39_wellord1,t7_wellord2,d1_wellord2,d5_funct_1,t10_wellord2,d2_ordinal1,t8_wellord2,t8_wellord2,t35_wellord1,t37_wellord1,t29_wellord1,t62_wellord1,d4_relat_2,d8_funct_1]), [file(wellord2,t13_wellord2),interesting(0.53)]). fof(t28_wellord2,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ~ ( ! [B] : ~ ( r2_hidden(B,A) & B = k1_xboole_0 ) & ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ~ ( k1_relat_1(B) = A & ! [C] : ( r2_hidden(C,A) => r2_hidden(k1_funct_1(B,C),C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t26_wellord2,d5_wellord1,d5_wellord1,d5_wellord1,d5_wellord1,d4_tarski,d4_relat_2,t92_zfmisc_1,d3_wellord1,d7_xboole_0,d6_relat_2,d1_relat_2,d3_xboole_0,d1_wellord1,s2_funct_1]), [file(wellord2,t28_wellord2),interesting(0.46)]). fof(t27_wellord2,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ~ ( ! [B] : ~ ( r2_hidden(B,A) & B = k1_xboole_0 ) & ! [B,C] : ( ( r2_hidden(B,A) & r2_hidden(C,A) ) => ( B = C | r1_xboole_0(B,C) ) ) & ! [B] : ? [C] : ( r2_hidden(C,A) & ! [D] : k3_xboole_0(B,C) != k1_tarski(D) ) ) ) ), inference(mizar_proof,[status(thm)],[t26_wellord2,d5_wellord1,d5_wellord1,d5_wellord1,d5_wellord1,d4_tarski,d4_relat_2,t92_zfmisc_1,d3_wellord1,d6_relat_2,d1_relat_2,t3_xboole_0,d1_wellord1,s2_funct_1,d3_tarski,d3_xboole_0,d5_funct_1,d1_tarski,d7_xboole_0,d3_xboole_0,d10_xboole_0,d3_tarski,d1_tarski,d5_funct_1,d3_xboole_0]), [file(wellord2,t27_wellord2),interesting(0.32)]). fof(s1_wellord2,theorem, ( ! [A] : ~ ( r2_hidden(A,f1_s1_wellord2) & ! [B] : ~ ( r2_hidden(B,f2_s1_wellord2) & p1_s1_wellord2(A,B) ) ) => ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s1_wellord2 & r1_tarski(k2_relat_1(A),f2_s1_wellord2) & ! [B] : ( r2_hidden(B,f1_s1_wellord2) => p1_s1_wellord2(B,k1_funct_1(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[s2_xboole_0,s1_xboole_0,s2_funct_1,t60_relat_1,t2_xboole_1,t60_relat_1,t65_relat_1,d5_funct_1,t28_wellord2,t46_relat_1,t47_relat_1,t46_relat_1,d3_tarski,d5_funct_1,d5_funct_1,t47_relat_1,t22_funct_1,d5_funct_1]), [file(wellord2,s1_wellord2),interesting(0.22)]). 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(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(s2_xboole_0,theorem, ( ( ! [A] : ( r2_hidden(A,f1_s2_xboole_0) <=> p1_s2_xboole_0(A) ) & ! [A] : ( r2_hidden(A,f2_s2_xboole_0) <=> p1_s2_xboole_0(A) ) ) => f1_s2_xboole_0 = f2_s2_xboole_0 ), file(xboole_0,s2_xboole_0), [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(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(t60_relat_1,theorem, ( k1_relat_1(k1_xboole_0) = k1_xboole_0 & k2_relat_1(k1_xboole_0) = k1_xboole_0 ), file(relat_1,t60_relat_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(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(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(t136_zfmisc_1,theorem,( ! [A] : ? [B] : ( r2_hidden(A,B) & ! [C,D] : ( ( r2_hidden(C,B) & r1_tarski(D,C) ) => r2_hidden(D,B) ) & ! [C] : ( r2_hidden(C,B) => r2_hidden(k1_zfmisc_1(C),B) ) & ! [C] : ~ ( r1_tarski(C,B) & ~ r2_tarski(C,B) & ~ r2_hidden(C,B) ) ) ), file(zfmisc_1,t136_zfmisc_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(d2_ordinal1,definition,( ! [A] : ( v1_ordinal1(A) <=> ! [B] : ( r2_hidden(B,A) => r1_tarski(B,A) ) ) ), file(ordinal1,d2_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(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(d1_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( r1_relat_2(A,B) <=> ! [C] : ( r2_hidden(C,B) => r2_hidden(k4_tarski(C,C),A) ) ) ) ), file(relat_2,d1_relat_2), [interesting(0.00)]). fof(d9_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ( v1_relat_2(A) <=> r1_relat_2(A,k3_relat_1(A)) ) ) ), file(relat_2,d9_relat_2), [interesting(0.00)]). fof(d8_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( r8_relat_2(A,B) <=> ! [C,D,E] : ( ( r2_hidden(C,B) & r2_hidden(D,B) & r2_hidden(E,B) & r2_hidden(k4_tarski(C,D),A) & r2_hidden(k4_tarski(D,E),A) ) => r2_hidden(k4_tarski(C,E),A) ) ) ) ), file(relat_2,d8_relat_2), [interesting(0.00)]). fof(d16_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ( v8_relat_2(A) <=> r8_relat_2(A,k3_relat_1(A)) ) ) ), file(relat_2,d16_relat_2), [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(d6_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( r6_relat_2(A,B) <=> ! [C,D] : ~ ( r2_hidden(C,B) & r2_hidden(D,B) & C != D & ~ r2_hidden(k4_tarski(C,D),A) & ~ r2_hidden(k4_tarski(D,C),A) ) ) ) ), file(relat_2,d6_relat_2), [interesting(0.00)]). fof(d14_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ( v6_relat_2(A) <=> r6_relat_2(A,k3_relat_1(A)) ) ) ), file(relat_2,d14_relat_2), [interesting(0.00)]). fof(d4_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( r4_relat_2(A,B) <=> ! [C,D] : ( ( r2_hidden(C,B) & r2_hidden(D,B) & r2_hidden(k4_tarski(C,D),A) & r2_hidden(k4_tarski(D,C),A) ) => C = D ) ) ) ), file(relat_2,d4_relat_2), [interesting(0.00)]). fof(d12_relat_2,definition,( ! [A] : ( v1_relat_1(A) => ( v4_relat_2(A) <=> r4_relat_2(A,k3_relat_1(A)) ) ) ), file(relat_2,d12_relat_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(d2_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ( v1_wellord1(A) <=> ! [B] : ~ ( r1_tarski(B,k3_relat_1(A)) & B != k1_xboole_0 & ! [C] : ~ ( r2_hidden(C,B) & r1_xboole_0(k1_wellord1(A,C),B) ) ) ) ) ), file(wellord1,d2_wellord1), [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(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(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(d1_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ! [B,C] : ( C = k1_wellord1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D != B & r2_hidden(k4_tarski(D,B),A) ) ) ) ) ), file(wellord1,d1_wellord1), [interesting(0.00)]). fof(d4_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ( v2_wellord1(A) <=> ( v1_relat_2(A) & v8_relat_2(A) & v4_relat_2(A) & v6_relat_2(A) & v1_wellord1(A) ) ) ) ), file(wellord1,d4_wellord1), [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(s1_relat_1,theorem,( ? [A] : ( v1_relat_1(A) & ! [B,C] : ( r2_hidden(k4_tarski(B,C),A) <=> ( r2_hidden(B,f1_s1_relat_1) & r2_hidden(C,f2_s1_relat_1) & p1_s1_relat_1(B,C) ) ) ) ), file(relat_1,s1_relat_1), [interesting(0.00)]). fof(d4_relat_1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( B = k1_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : r2_hidden(k4_tarski(C,D),A) ) ) ) ), file(relat_1,d4_relat_1), [interesting(0.00)]). fof(d5_relat_1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : r2_hidden(k4_tarski(D,C),A) ) ) ) ), file(relat_1,d5_relat_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(t30_relat_1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( r2_hidden(k4_tarski(A,B),C) => ( r2_hidden(A,k3_relat_1(C)) & r2_hidden(B,k3_relat_1(C)) ) ) ) ), file(relat_1,t30_relat_1), [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(t49_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r3_wellord1(A,B,C) => r3_wellord1(B,A,k2_funct_1(C)) ) ) ) ) ), file(wellord1,t49_wellord1), [interesting(0.00)]). fof(t54_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( v2_wellord1(A) & r3_wellord1(A,B,C) ) => v2_wellord1(B) ) ) ) ) ), file(wellord1,t54_wellord1), [interesting(0.00)]). fof(t8_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ( r2_wellord1(A,k3_relat_1(A)) <=> v2_wellord1(A) ) ) ), file(wellord1,t8_wellord1), [interesting(0.00)]). fof(d5_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( r2_wellord1(A,B) <=> ( r1_relat_2(A,B) & r8_relat_2(A,B) & r4_relat_2(A,B) & r6_relat_2(A,B) & r1_wellord1(A,B) ) ) ) ), file(wellord1,d5_wellord1), [interesting(0.00)]). fof(t20_wellord1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( r1_tarski(k3_relat_1(k2_wellord1(B,A)),k3_relat_1(B)) & r1_tarski(k3_relat_1(k2_wellord1(B,A)),A) ) ) ), file(wellord1,t20_wellord1), [interesting(0.00)]). fof(t16_wellord1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( r2_hidden(A,k2_wellord1(C,B)) <=> ( r2_hidden(A,C) & r2_hidden(A,k2_zfmisc_1(B,B)) ) ) ) ), file(wellord1,t16_wellord1), [interesting(0.00)]). fof(d3_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( r1_wellord1(A,B) <=> ! [C] : ~ ( r1_tarski(C,B) & C != k1_xboole_0 & ! [D] : ~ ( r2_hidden(D,C) & r1_xboole_0(k1_wellord1(A,D),C) ) ) ) ) ), file(wellord1,d3_wellord1), [interesting(0.00)]). fof(t3_xboole_0,theorem,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ ( r2_hidden(C,A) & r2_hidden(C,B) ) ) & ~ ( ? [C] : ( r2_hidden(C,A) & r2_hidden(C,B) ) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t3_xboole_0), [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(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [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(t6_zfmisc_1,theorem,( ! [A,B] : ( r1_tarski(k1_tarski(A),k1_tarski(B)) => A = B ) ), file(zfmisc_1,t6_zfmisc_1), [interesting(0.00)]). fof(t32_wellord1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( v2_wellord1(B) => v2_wellord1(k2_wellord1(B,A)) ) ) ), file(wellord1,t32_wellord1), [interesting(0.00)]). fof(t39_wellord1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( ( v2_wellord1(B) & r1_tarski(A,k3_relat_1(B)) ) => k3_relat_1(k2_wellord1(B,A)) = A ) ) ), file(wellord1,t39_wellord1), [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(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(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(t47_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(k1_relat_1(A),k2_relat_1(B)) => k2_relat_1(k5_relat_1(B,A)) = k2_relat_1(A) ) ) ) ), file(relat_1,t47_relat_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(s2_wellord2,theorem, ( ! [A] : ~ ( r2_hidden(A,f1_s2_wellord2) & ! [B,C] : ~ ( r2_hidden(B,f2_s2_wellord2) & r2_hidden(C,f3_s2_wellord2) & p1_s2_wellord2(A,B,C) ) ) => ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ? [B] : ( v1_relat_1(B) & v1_funct_1(B) & k1_relat_1(A) = f1_s2_wellord2 & k1_relat_1(B) = f1_s2_wellord2 & ! [C] : ( r2_hidden(C,f1_s2_wellord2) => p1_s2_wellord2(C,k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ), inference(mizar_proof,[status(thm)],[t106_zfmisc_1,t7_mcart_1,s1_wellord2,s3_funct_1,s3_funct_1]), [file(wellord2,s2_wellord2),interesting(0.00)]). fof(t19_wellord1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( r2_hidden(A,k3_relat_1(k2_wellord1(C,B))) => ( r2_hidden(A,k3_relat_1(C)) & r2_hidden(A,B) ) ) ) ), file(wellord1,t19_wellord1), [interesting(0.00)]). fof(t35_wellord1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( ( v2_wellord1(C) & r2_hidden(A,k3_relat_1(C)) & r2_hidden(B,k1_wellord1(C,A)) ) => k1_wellord1(k2_wellord1(C,k1_wellord1(C,A)),B) = k1_wellord1(C,B) ) ) ), file(wellord1,t35_wellord1), [interesting(0.00)]). fof(t37_wellord1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( ( v2_wellord1(C) & r2_hidden(A,k3_relat_1(C)) & r2_hidden(B,k3_relat_1(C)) ) => ( r2_hidden(k4_tarski(A,B),C) <=> r1_tarski(k1_wellord1(C,A),k1_wellord1(C,B)) ) ) ) ), file(wellord1,t37_wellord1), [interesting(0.00)]). fof(t29_wellord1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( r1_tarski(A,B) => k2_wellord1(k2_wellord1(C,B),A) = k2_wellord1(C,A) ) ) ), file(wellord1,t29_wellord1), [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(t52_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ! [C] : ( v1_relat_1(C) => ( ( r4_wellord1(A,B) & r4_wellord1(B,C) ) => r4_wellord1(A,C) ) ) ) ) ), file(wellord1,t52_wellord1), [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(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(t13_wellord1,theorem,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k1_wellord1(B,A),k3_relat_1(B)) ) ), file(wellord1,t13_wellord1), [interesting(0.00)]). fof(d2_relat_1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( A = B <=> ! [C,D] : ( r2_hidden(k4_tarski(C,D),A) <=> r2_hidden(k4_tarski(C,D),B) ) ) ) ) ), file(relat_1,d2_relat_1), [interesting(0.00)]). fof(t57_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ( v2_wellord1(A) => ! [B] : ~ ( r2_hidden(B,k3_relat_1(A)) & r4_wellord1(A,k2_wellord1(A,k1_wellord1(A,B))) ) ) ) ), file(wellord1,t57_wellord1), [interesting(0.00)]). fof(t36_ordinal1,theorem,( ! [A] : ~ ( ! [B] : ( r2_hidden(B,A) => v3_ordinal1(B) ) & ! [B] : ( v3_ordinal1(B) => ~ r1_tarski(A,B) ) ) ), file(ordinal1,t36_ordinal1), [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(t61_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( v2_wellord1(A) & r3_wellord1(A,B,C) ) => ! [D] : ~ ( r2_hidden(D,k3_relat_1(A)) & ! [E] : ~ ( r2_hidden(E,k3_relat_1(B)) & r4_wellord1(k2_wellord1(A,k1_wellord1(A,D)),k2_wellord1(B,k1_wellord1(B,E))) ) ) ) ) ) ) ), file(wellord1,t61_wellord1), [interesting(0.00)]). fof(t36_wellord1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( ( v2_wellord1(B) & r1_tarski(A,k3_relat_1(B)) ) => ( ~ ( A != k3_relat_1(B) & ! [C] : ~ ( r2_hidden(C,k3_relat_1(B)) & A = k1_wellord1(B,C) ) ) <=> ! [C] : ( r2_hidden(C,A) => ! [D] : ( r2_hidden(k4_tarski(D,C),B) => r2_hidden(D,A) ) ) ) ) ) ), file(wellord1,t36_wellord1), [interesting(0.00)]). fof(d8_wellord1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r4_wellord1(A,B) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & r3_wellord1(A,B,C) ) ) ) ) ), file(wellord1,d8_wellord1), [interesting(0.00)]). fof(t58_wellord1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ~ ( v2_wellord1(C) & r2_hidden(A,k3_relat_1(C)) & r2_hidden(B,k3_relat_1(C)) & A != B & r4_wellord1(k2_wellord1(C,k1_wellord1(C,A)),k2_wellord1(C,k1_wellord1(C,B))) ) ) ), file(wellord1,t58_wellord1), [interesting(0.00)]). fof(t62_wellord1,theorem,( ! [A,B,C,D] : ( v1_relat_1(D) => ! [E] : ( v1_relat_1(E) => ( ( v2_wellord1(D) & v2_wellord1(E) & r2_hidden(A,k3_relat_1(D)) & r2_hidden(B,k3_relat_1(E)) & r2_hidden(C,k3_relat_1(E)) & r4_wellord1(D,k2_wellord1(E,k1_wellord1(E,B))) & r4_wellord1(k2_wellord1(D,k1_wellord1(D,A)),k2_wellord1(E,k1_wellord1(E,C))) ) => ( r1_tarski(k1_wellord1(E,C),k1_wellord1(E,B)) & r2_hidden(k4_tarski(C,B),E) ) ) ) ) ), file(wellord1,t62_wellord1), [interesting(0.00)]). fof(t41_wellord1,theorem,( ! [A] : ( v1_relat_1(A) => ( v2_wellord1(A) => ! [B] : ( ! [C] : ( ( r2_hidden(C,k3_relat_1(A)) & r1_tarski(k1_wellord1(A,C),B) ) => r2_hidden(C,B) ) => r1_tarski(k3_relat_1(A),B) ) ) ) ), file(wellord1,t41_wellord1), [interesting(0.00)]). fof(t15_wellord2,theorem,( $true ), file(wellord2,t15_wellord2), [interesting(0.00)]). fof(t16_wellord2,theorem,( $true ), file(wellord2,t16_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(t64_wellord1,theorem,( ! [A,B] : ( v1_relat_1(B) => ~ ( r1_tarski(A,k3_relat_1(B)) & v2_wellord1(B) & ~ r4_wellord1(B,k2_wellord1(B,A)) & ! [C] : ~ ( r2_hidden(C,k3_relat_1(B)) & r4_wellord1(k2_wellord1(B,k1_wellord1(B,C)),k2_wellord1(B,A)) ) ) ) ), file(wellord1,t64_wellord1), [interesting(0.00)]). fof(t18_wellord2,theorem,( $true ), file(wellord2,t18_wellord2), [interesting(0.00)]). fof(t19_wellord2,theorem,( $true ), file(wellord2,t19_wellord2), [interesting(0.00)]). fof(t1_wellord2,theorem,( $true ), file(wellord2,t1_wellord2), [interesting(0.00)]). fof(t20_wellord2,theorem,( $true ), file(wellord2,t20_wellord2), [interesting(0.00)]). fof(t21_wellord2,theorem,( $true ), file(wellord2,t21_wellord2), [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(t23_wellord2,theorem,( $true ), file(wellord2,t23_wellord2), [interesting(0.00)]). fof(t24_wellord2,theorem,( $true ), file(wellord2,t24_wellord2), [interesting(0.00)]).