% Mizar problem: t29_zf_refle,zf_refle,656,50 fof(t29_zf_refle,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) ) => ( ( r2_hidden(k5_ordinal2,A) & ! [C] : ( m1_ordinal4(C,A) => ! [D] : ( m1_ordinal4(D,A) => ( r2_hidden(C,D) => r1_tarski(k5_zf_refle(A,B,C),k5_zf_refle(A,B,D)) ) ) ) & ! [C] : ( m1_ordinal4(C,A) => ( v4_ordinal1(C) => ( C = k1_xboole_0 | k5_zf_refle(A,B,C) = k3_card_3(k2_ordinal1(B,C)) ) ) ) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ? [D] : ( m2_ordinal4(D,A) & v2_ordinal2(D) & v3_ordinal2(D) & ! [E] : ( m1_ordinal4(E,A) => ( k4_ordinal4(A,D,E) = E => ( k1_xboole_0 = E | ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k1_zf_lang,k5_zf_refle(A,B,E)) & m2_relset_1(F,k1_zf_lang,k5_zf_refle(A,B,E)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,E),k4_zf_refle(A,B),F),C) <=> r1_zf_model(k5_zf_refle(A,B,E),F,C) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,antisymmetry_r2_xboole_0,cc1_card_1,cc1_classes2,cc1_funct_1,cc1_ordinal1,cc1_ordinal2,cc1_relset_1,cc2_card_1,cc2_classes2,cc2_funct_1,cc2_ordinal1,cc3_card_1,cc3_classes2,cc3_ordinal1,commutativity_k2_xboole_0,commutativity_k3_xboole_0,commutativity_k3_zf_refle,connectedness_r1_ordinal1,connectedness_r1_xreal_0,d10_xboole_0,d12_funct_1,d15_zf_lang,d16_ordinal2,d17_ordinal2,d1_funct_2,d1_zf_lang,d1_zf_lang1,d1_zf_refle,d2_funct_2,d2_ordinal1,d2_ordinal2,d2_ordinal4,d2_zf_lang1,d2_zf_refle,d30_zf_lang,d3_ordinal4,d3_tarski,d3_xboole_0,d3_zf_lang,d4_card_3,d4_ordinal2,d4_tarski,d4_zf_lang,d5_funct_1,d5_zf_lang,d5_zf_refle,d6_zf_lang,d7_ordinal1,d7_zf_lang,d8_ordinal2,d8_xboole_0,d9_ordinal2,dt_k12_finseq_1,dt_k12_ordinal2,dt_k18_zf_lang,dt_k19_zf_lang,dt_k1_card_1,dt_k1_fraenkel,dt_k1_funct_1,dt_k1_funct_2,dt_k1_numbers,dt_k1_ordinal1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zf_lang,dt_k1_zf_lang1,dt_k1_zf_model,dt_k1_zf_refle,dt_k1_zfmisc_1,dt_k20_zf_lang,dt_k21_zf_lang,dt_k22_zf_lang,dt_k23_zf_lang,dt_k24_zf_lang,dt_k2_ordinal1,dt_k2_ordinal2,dt_k2_ordinal4,dt_k2_relat_1,dt_k2_xboole_0,dt_k2_zf_lang1,dt_k2_zf_model,dt_k2_zf_refle,dt_k2_zfmisc_1,dt_k3_card_3,dt_k3_ordinal4,dt_k3_tarski,dt_k3_xboole_0,dt_k3_zf_lang,dt_k3_zf_refle,dt_k4_classes1,dt_k4_ordinal2,dt_k4_ordinal4,dt_k4_relset_1,dt_k4_zf_lang,dt_k4_zf_refle,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_ordinal4,dt_k5_relat_1,dt_k5_zf_lang,dt_k5_zf_refle,dt_k6_ordinal4,dt_k6_relat_1,dt_k6_zf_lang,dt_k7_finseq_1,dt_k7_ordinal2,dt_k7_relat_1,dt_k7_zf_lang,dt_k8_finseq_1,dt_k8_funct_2,dt_k8_ordinal2,dt_k8_relat_1,dt_k8_zf_lang,dt_k9_relat_1,dt_m1_finseq_1,dt_m1_fraenkel,dt_m1_ordinal1,dt_m1_ordinal4,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_fraenkel,dt_m2_ordinal4,dt_m2_relset_1,dt_m2_subset_1,existence_m1_finseq_1,existence_m1_fraenkel,existence_m1_ordinal1,existence_m1_ordinal4,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_fraenkel,existence_m2_ordinal4,existence_m2_relset_1,existence_m2_subset_1,fc1_funct_1,fc1_ordinal1,fc1_ordinal2,fc1_subset_1,fc1_zf_lang,fc1_zf_lang1,fc2_card_1,fc2_classes2,fc2_funct_1,fc2_ordinal1,fc2_ordinal2,fc2_zf_lang,fc3_ordinal1,fc3_ordinal2,fc3_zf_lang,fc4_funct_1,fc4_ordinal1,fc4_subset_1,fc4_zf_lang,fc5_funct_1,fc5_ordinal1,fc5_zf_lang,fc6_funct_1,fc6_zf_lang,fc7_classes2,fraenkel_a_0_0_zf_lang,idempotence_k2_xboole_0,idempotence_k3_xboole_0,idempotence_k3_zf_refle,irreflexivity_r2_xboole_0,rc1_card_1,rc1_classes2,rc1_funct_1,rc1_ordinal1,rc1_ordinal2,rc1_subset_1,rc1_zf_lang,rc1_zf_refle,rc2_card_1,rc2_classes2,rc2_funct_1,rc2_ordinal1,rc2_ordinal2,rc2_subset_1,rc2_zf_refle,rc3_funct_1,rc3_ordinal1,rc4_funct_1,rc4_ordinal1,rc5_funct_1,redefinition_k12_finseq_1,redefinition_k1_fraenkel,redefinition_k2_ordinal1,redefinition_k2_zf_model,redefinition_k2_zf_refle,redefinition_k3_zf_lang,redefinition_k3_zf_refle,redefinition_k4_ordinal4,redefinition_k4_relset_1,redefinition_k4_zf_refle,redefinition_k5_numbers,redefinition_k5_ordinal4,redefinition_k5_zf_refle,redefinition_k6_ordinal4,redefinition_k8_finseq_1,redefinition_k8_funct_2,redefinition_m2_finseq_1,redefinition_m2_fraenkel,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_ordinal1,redefinition_r2_wellord2,reflexivity_r1_ordinal1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r2_wellord2,s1_partfun1__e18_36_4_6__zf_refle,s1_zf_lang__e8_36__zf_refle,s2_zf_refle__e8_36_4__zf_refle,s3_card_3__e5_36_4_6_1__zf_refle,s3_zf_refle__e13_36_4__zf_refle,s4_zf_refle__e15_36_4__zf_refle,s5_zf_refle__e26_36_4__zf_refle,s5_zf_refle__e7_36_4_4__zf_refle,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,spc4_boole,spc4_numerals,spc5_boole,spc5_numerals,symmetry_r2_wellord2,t10_ordinal1,t10_ordinal3,t10_ordinal4,t10_zfmodel1,t11_funct_2,t11_relset_1,t12_zf_model,t13_ordinal4,t13_zf_model,t14_zf_model,t15_ordinal4,t15_zf_model,t16_zf_model,t17_ordinal4,t17_zf_refle,t19_ordinal1,t1_boole,t1_classes2,t1_numerals,t1_subset,t1_xboole_1,t21_card_1,t21_ordinal1,t22_funct_1,t22_ordinal1,t23_ordinal1,t24_ordinal1,t24_zf_refle,t25_zf_refle,t26_ordinal2,t27_ordinal2,t28_xboole_1,t28_zf_refle,t29_ordinal2,t2_boole,t2_classes1,t2_subset,t2_tarski,t34_ordinal1,t35_funct_1,t35_ordinal4,t36_ordinal4,t3_card_2,t3_subset,t41_ordinal1,t43_funct_1,t4_subset,t53_zf_lang,t54_zf_lang,t58_zf_lang,t5_subset,t62_zf_lang,t63_funct_5,t67_classes2,t68_funct_5,t6_boole,t70_funct_1,t71_relat_1,t72_funct_1,t73_funct_1,t7_boole,t7_ordinal1,t7_xboole_1,t80_zf_lang1,t84_zf_lang1,t87_relat_1,t8_boole,t8_ordinal4,t8_zf_refle,t90_relat_1,t91_relat_1,t94_relat_1]), [file(zf_refle,t29_zf_refle)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(antisymmetry_r2_xboole_0,axiom,( ! [A,B] : ( r2_xboole_0(A,B) => ~ r2_xboole_0(B,A) ) ), file(xboole_0,r2_xboole_0), []). fof(cc1_card_1,axiom,( ! [A] : ( v1_card_1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(card_1,cc1_card_1), []). fof(cc1_classes2,axiom,( ! [A] : ( v2_classes1(A) => v1_classes1(A) ) ), file(classes2,cc1_classes2), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) ) ) ), file(ordinal1,cc1_ordinal1), []). fof(cc1_ordinal2,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_ordinal1(B,A) => v1_ordinal2(B) ) ) ), file(ordinal2,cc1_ordinal2), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc2_card_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc2_card_1), []). fof(cc2_classes2,axiom,( ! [A] : ( v1_classes2(A) => ( v1_ordinal1(A) & v1_classes1(A) & v2_classes1(A) ) ) ), file(classes2,cc2_classes2), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_ordinal1,axiom,( ! [A] : ( ( v1_ordinal1(A) & v2_ordinal1(A) ) => v3_ordinal1(A) ) ), file(ordinal1,cc2_ordinal1), []). fof(cc3_card_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_finset_1(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc3_card_1), []). fof(cc3_classes2,axiom,( ! [A] : ( ( v1_ordinal1(A) & v2_classes1(A) ) => v1_classes2(A) ) ), file(classes2,cc3_classes2), []). fof(cc3_ordinal1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(ordinal1,cc3_ordinal1), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). fof(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k3_zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_ordinal4(B,A) & m1_ordinal4(C,A) ) => k3_zf_refle(A,B,C) = k3_zf_refle(A,C,B) ) ), file(zf_refle,k3_zf_refle), []). fof(connectedness_r1_ordinal1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => ( r1_ordinal1(A,B) | r1_ordinal1(B,A) ) ) ), file(ordinal1,r1_ordinal1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d12_funct_1,axiom,( ! [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), []). fof(d15_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( v7_zf_lang(A) <=> ( v2_zf_lang(A) | v3_zf_lang(A) ) ) ) ), file(zf_lang,d15_zf_lang), []). fof(d16_ordinal2,axiom,( ! [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), []). fof(d17_ordinal2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ( v3_ordinal2(A) <=> ! [B] : ( v3_ordinal1(B) => ! [C] : ( v3_ordinal1(C) => ( ( r2_hidden(B,k1_relat_1(A)) & v4_ordinal1(B) & C = k1_funct_1(A,B) ) => ( B = k1_xboole_0 | r1_ordinal2(C,k2_ordinal1(A,B)) ) ) ) ) ) ) ), file(ordinal2,d17_ordinal2), []). fof(d1_funct_2,axiom,( ! [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), []). fof(d1_zf_lang,axiom,( k1_zf_lang = a_0_0_zf_lang ), file(zf_lang,d1_zf_lang), []). fof(d1_zf_lang1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m2_relset_1(B,k1_zf_lang,A) ) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ! [D] : ( m1_subset_1(D,A) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,A) & m2_relset_1(E,k1_zf_lang,A) ) => ( E = k1_zf_lang1(A,B,C,D) <=> ( k8_funct_2(k1_zf_lang,A,E,C) = D & ! [F] : ( m2_subset_1(F,k5_numbers,k1_zf_lang) => ( k8_funct_2(k1_zf_lang,A,E,F) != k8_funct_2(k1_zf_lang,A,B,F) => C = F ) ) ) ) ) ) ) ) ) ), file(zf_lang1,d1_zf_lang1), []). fof(d1_zf_refle,axiom,( ! [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), []). fof(d2_funct_2,axiom,( ! [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), []). fof(d2_ordinal1,axiom,( ! [A] : ( v1_ordinal1(A) <=> ! [B] : ( r2_hidden(B,A) => r1_tarski(B,A) ) ) ), file(ordinal1,d2_ordinal1), []). fof(d2_ordinal2,axiom,( ! [A,B] : ( B = k2_ordinal2(A) <=> ! [C] : ( r2_hidden(C,B) <=> ( r2_hidden(C,A) & v3_ordinal1(C) ) ) ) ), file(ordinal2,d2_ordinal2), []). fof(d2_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( v3_ordinal1(B) => ( m1_ordinal4(B,A) <=> r2_hidden(B,A) ) ) ) ), file(ordinal4,d2_ordinal4), []). fof(d2_zf_lang1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r1_tarski(B,C) => k2_zf_lang1(A,B,C,D) = D ) ) ) ) ) ), file(zf_lang1,d2_zf_lang1), []). fof(d2_zf_refle,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) ) => ! [C] : ( m1_ordinal4(C,B) => k1_zf_refle(A,B,C) = k3_card_3(k8_relat_1(B,k7_relat_1(A,k4_classes1(C)))) ) ) ) ), file(zf_refle,d2_zf_refle), []). fof(d30_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( v4_zf_lang(A) => ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => ( B = k20_zf_lang(A) <=> k6_zf_lang(B) = A ) ) ) ) ), file(zf_lang,d30_zf_lang), []). fof(d3_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_ordinal2(B) ) => ( m2_ordinal4(B,A) <=> ( k1_relat_1(B) = k2_ordinal2(A) & r1_tarski(k2_relat_1(B),k2_ordinal2(A)) ) ) ) ) ), file(ordinal4,d3_ordinal4), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d3_xboole_0,axiom,( ! [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), []). fof(d3_zf_lang,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => k4_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,0),k3_zf_lang(A)),k3_zf_lang(B)) ) ) ), file(zf_lang,d3_zf_lang), []). fof(d4_card_3,axiom,( ! [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), []). fof(d4_ordinal2,axiom,( k4_ordinal2 = k1_ordinal1(k1_xboole_0) ), file(ordinal2,d4_ordinal2), []). fof(d4_tarski,axiom,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), []). fof(d4_zf_lang,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => k5_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,1),k3_zf_lang(A)),k3_zf_lang(B)) ) ) ), file(zf_lang,d4_zf_lang), []). fof(d5_funct_1,axiom,( ! [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), []). fof(d5_zf_lang,axiom,( ! [A] : ( m2_finseq_1(A,k5_numbers) => k6_zf_lang(A) = k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,2),A) ) ), file(zf_lang,d5_zf_lang), []). fof(d5_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal1(B,A) => ( v1_zf_refle(B,A) <=> k1_relat_1(B) = k2_ordinal2(A) ) ) ) ), file(zf_refle,d5_zf_refle), []). fof(d6_zf_lang,axiom,( ! [A] : ( m2_finseq_1(A,k5_numbers) => ! [B] : ( m2_finseq_1(B,k5_numbers) => k7_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,3),A),B) ) ) ), file(zf_lang,d6_zf_lang), []). fof(d7_ordinal1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v5_ordinal1(A) <=> v3_ordinal1(k1_relat_1(A)) ) ) ), file(ordinal1,d7_ordinal1), []). fof(d7_zf_lang,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_finseq_1(B,k5_numbers) => k8_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,4),k3_zf_lang(A)),B) ) ) ), file(zf_lang,d7_zf_lang), []). fof(d8_ordinal2,axiom,( ! [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), []). fof(d8_xboole_0,axiom,( ! [A,B] : ( r2_xboole_0(A,B) <=> ( r1_tarski(A,B) & A != B ) ) ), file(xboole_0,d8_xboole_0), []). fof(d9_ordinal2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) ) => k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A)) ) ), file(ordinal2,d9_ordinal2), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k12_ordinal2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => v3_ordinal1(k12_ordinal2(A)) ) ), file(ordinal2,k12_ordinal2), []). fof(dt_k18_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => m2_subset_1(k18_zf_lang(A),k5_numbers,k1_zf_lang) ) ), file(zf_lang,k18_zf_lang), []). fof(dt_k19_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => m2_subset_1(k19_zf_lang(A),k5_numbers,k1_zf_lang) ) ), file(zf_lang,k19_zf_lang), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => m1_fraenkel(k1_fraenkel(A,B),A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_funct_2,axiom,( $true ), file(funct_2,k1_funct_2), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_ordinal1,axiom,( $true ), file(ordinal1,k1_ordinal1), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zf_lang,axiom,( m1_subset_1(k1_zf_lang,k1_zfmisc_1(k5_numbers)) ), file(zf_lang,k1_zf_lang), []). fof(dt_k1_zf_lang1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m1_relset_1(B,k1_zf_lang,A) & m1_subset_1(C,k1_zf_lang) & m1_subset_1(D,A) ) => ( v1_funct_1(k1_zf_lang1(A,B,C,D)) & v1_funct_2(k1_zf_lang1(A,B,C,D),k1_zf_lang,A) & m2_relset_1(k1_zf_lang1(A,B,C,D),k1_zf_lang,A) ) ) ), file(zf_lang1,k1_zf_lang1), []). fof(dt_k1_zf_model,axiom,( $true ), file(zf_model,k1_zf_model), []). fof(dt_k1_zf_refle,axiom,( $true ), file(zf_refle,k1_zf_refle), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k20_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_zf_lang(k20_zf_lang(A)) & m2_finseq_1(k20_zf_lang(A),k5_numbers) ) ) ), file(zf_lang,k20_zf_lang), []). fof(dt_k21_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_zf_lang(k21_zf_lang(A)) & m2_finseq_1(k21_zf_lang(A),k5_numbers) ) ) ), file(zf_lang,k21_zf_lang), []). fof(dt_k22_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_zf_lang(k22_zf_lang(A)) & m2_finseq_1(k22_zf_lang(A),k5_numbers) ) ) ), file(zf_lang,k22_zf_lang), []). fof(dt_k23_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => m2_subset_1(k23_zf_lang(A),k5_numbers,k1_zf_lang) ) ), file(zf_lang,k23_zf_lang), []). fof(dt_k24_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_zf_lang(k24_zf_lang(A)) & m2_finseq_1(k24_zf_lang(A),k5_numbers) ) ) ), file(zf_lang,k24_zf_lang), []). fof(dt_k2_ordinal1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v3_ordinal1(B) ) => m1_ordinal1(k2_ordinal1(A,B),k2_relat_1(A)) ) ), file(ordinal1,k2_ordinal1), []). fof(dt_k2_ordinal2,axiom,( $true ), file(ordinal2,k2_ordinal2), []). fof(dt_k2_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => m1_ordinal4(k2_ordinal4(A),A) ) ), file(ordinal4,k2_ordinal4), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zf_lang1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) ) => ( v1_funct_1(k2_zf_lang1(A,B,C,D)) & v1_funct_2(k2_zf_lang1(A,B,C,D),A,C) & m2_relset_1(k2_zf_lang1(A,B,C,D),A,C) ) ) ), file(zf_lang1,k2_zf_lang1), []). fof(dt_k2_zf_model,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => m1_subset_1(k2_zf_model(A),k1_zfmisc_1(k1_zf_lang)) ) ), file(zf_model,k2_zf_model), []). fof(dt_k2_zf_refle,axiom,( ! [A,B,C] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) & ~ v1_xboole_0(B) & v1_classes2(B) & m1_ordinal4(C,B) ) => m1_ordinal4(k2_zf_refle(A,B,C),B) ) ), file(zf_refle,k2_zf_refle), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_card_3,axiom,( $true ), file(card_3,k3_card_3), []). fof(dt_k3_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ( ~ v1_xboole_0(k3_ordinal4(A)) & m1_ordinal4(k3_ordinal4(A),A) ) ) ), file(ordinal4,k3_ordinal4), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k3_zf_lang,axiom,( ! [A] : ( m1_subset_1(A,k1_zf_lang) => m2_finseq_1(k3_zf_lang(A),k5_numbers) ) ), file(zf_lang,k3_zf_lang), []). fof(dt_k3_zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_ordinal4(B,A) & m1_ordinal4(C,A) ) => m1_ordinal4(k3_zf_refle(A,B,C),A) ) ), file(zf_refle,k3_zf_refle), []). fof(dt_k4_classes1,axiom,( $true ), file(classes1,k4_classes1), []). fof(dt_k4_ordinal2,axiom, ( v3_ordinal1(k4_ordinal2) & ~ v1_xboole_0(k4_ordinal2) ), file(ordinal2,k4_ordinal2), []). fof(dt_k4_ordinal4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) & m1_ordinal4(C,A) ) => m1_ordinal4(k4_ordinal4(A,B,C),A) ) ), file(ordinal4,k4_ordinal4), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k4_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => m2_finseq_1(k4_zf_lang(A,B),k5_numbers) ) ), file(zf_lang,k4_zf_lang), []). fof(dt_k4_zf_refle,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) ) => ( ~ v1_xboole_0(k4_zf_refle(A,B)) & m1_subset_1(k4_zf_refle(A,B),k1_zfmisc_1(A)) ) ) ), file(zf_refle,k4_zf_refle), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_ordinal4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) & m2_ordinal4(C,A) ) => m2_ordinal4(k5_ordinal4(A,B,C),A) ) ), file(ordinal4,k5_ordinal4), []). fof(dt_k5_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_1), []). fof(dt_k5_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => m2_finseq_1(k5_zf_lang(A,B),k5_numbers) ) ), file(zf_lang,k5_zf_lang), []). fof(dt_k5_zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) & m1_ordinal4(C,A) ) => ( ~ v1_xboole_0(k5_zf_refle(A,B,C)) & m1_subset_1(k5_zf_refle(A,B,C),A) ) ) ), file(zf_refle,k5_zf_refle), []). fof(dt_k6_ordinal4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_ordinal4(B,A) ) => ( ~ v1_xboole_0(k6_ordinal4(A,B)) & m1_ordinal4(k6_ordinal4(A,B),A) ) ) ), file(ordinal4,k6_ordinal4), []). fof(dt_k6_relat_1,axiom,( ! [A] : v1_relat_1(k6_relat_1(A)) ), file(relat_1,k6_relat_1), []). fof(dt_k6_zf_lang,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => m2_finseq_1(k6_zf_lang(A),k5_numbers) ) ), file(zf_lang,k6_zf_lang), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k7_ordinal2,axiom,( ! [A] : v3_ordinal1(k7_ordinal2(A)) ), file(ordinal2,k7_ordinal2), []). fof(dt_k7_relat_1,axiom,( ! [A,B] : ( v1_relat_1(A) => v1_relat_1(k7_relat_1(A,B)) ) ), file(relat_1,k7_relat_1), []). fof(dt_k7_zf_lang,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k5_numbers) & m1_finseq_1(B,k5_numbers) ) => m2_finseq_1(k7_zf_lang(A,B),k5_numbers) ) ), file(zf_lang,k7_zf_lang), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_ordinal2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) ) => v3_ordinal1(k8_ordinal2(A)) ) ), file(ordinal2,k8_ordinal2), []). fof(dt_k8_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => v1_relat_1(k8_relat_1(A,B)) ) ), file(relat_1,k8_relat_1), []). fof(dt_k8_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_finseq_1(B,k5_numbers) ) => m2_finseq_1(k8_zf_lang(A,B),k5_numbers) ) ), file(zf_lang,k8_zf_lang), []). fof(dt_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_1), []). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_fraenkel,axiom,( ! [A,B,C] : ( m1_fraenkel(C,A,B) => ( ~ v1_xboole_0(C) & v1_fraenkel(C) ) ) ), file(fraenkel,m1_fraenkel), []). fof(dt_m1_ordinal1,axiom,( ! [A,B] : ( m1_ordinal1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) ) ) ), file(ordinal1,m1_ordinal1), []). fof(dt_m1_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal4(B,A) => v3_ordinal1(B) ) ) ), file(ordinal4,m1_ordinal4), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ! [D] : ( m2_fraenkel(D,A,B,C) => ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) ) ) ), file(fraenkel,m2_fraenkel), []). fof(dt_m2_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m2_ordinal4(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_ordinal2(B) ) ) ) ), file(ordinal4,m2_ordinal4), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_fraenkel,axiom,( ! [A,B] : ? [C] : m1_fraenkel(C,A,B) ), file(fraenkel,m1_fraenkel), []). fof(existence_m1_ordinal1,axiom,( ! [A] : ? [B] : m1_ordinal1(B,A) ), file(ordinal1,m1_ordinal1), []). fof(existence_m1_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ? [B] : m1_ordinal4(B,A) ) ), file(ordinal4,m1_ordinal4), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ? [D] : m2_fraenkel(D,A,B,C) ) ), file(fraenkel,m2_fraenkel), []). fof(existence_m2_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ? [B] : m2_ordinal4(B,A) ) ), file(ordinal4,m2_ordinal4), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), []). fof(fc1_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k5_relat_1(A,B)) & v1_funct_1(k5_relat_1(A,B)) ) ) ), file(funct_1,fc1_funct_1), []). fof(fc1_ordinal1,axiom,( ! [A] : ~ v1_xboole_0(k1_ordinal1(A)) ), file(ordinal1,fc1_ordinal1), []). fof(fc1_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_zf_lang,axiom,( ~ v1_xboole_0(k1_zf_lang) ), file(zf_lang,fc1_zf_lang), []). fof(fc1_zf_lang1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => v1_finset_1(k1_zf_model(A)) ) ), file(zf_lang1,fc1_zf_lang1), []). fof(fc2_card_1,axiom,( ! [A] : ( v1_finset_1(A) => ( v1_ordinal1(k1_card_1(A)) & v2_ordinal1(k1_card_1(A)) & v3_ordinal1(k1_card_1(A)) & v1_finset_1(k1_card_1(A)) & v1_card_1(k1_card_1(A)) ) ) ), file(card_1,fc2_card_1), []). fof(fc2_classes2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ( v1_ordinal1(k2_ordinal2(A)) & v2_ordinal1(k2_ordinal2(A)) & v3_ordinal1(k2_ordinal2(A)) ) ) ), file(classes2,fc2_classes2), []). fof(fc2_funct_1,axiom,( ! [A] : ( v1_relat_1(k6_relat_1(A)) & v1_funct_1(k6_relat_1(A)) ) ), file(funct_1,fc2_funct_1), []). fof(fc2_ordinal1,axiom, ( v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_xboole_0(k1_xboole_0) & v1_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) ), file(ordinal1,fc2_ordinal1), []). fof(fc2_ordinal2,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) & v3_ordinal1(B) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) & v1_ordinal2(k7_relat_1(A,B)) ) ) ), file(ordinal2,fc2_ordinal2), []). fof(fc2_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => ( v1_relat_1(k4_zf_lang(A,B)) & v1_funct_1(k4_zf_lang(A,B)) & v1_finset_1(k4_zf_lang(A,B)) & v1_finseq_1(k4_zf_lang(A,B)) & v1_zf_lang(k4_zf_lang(A,B)) ) ) ), file(zf_lang,fc2_zf_lang), []). fof(fc3_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ( ~ v1_xboole_0(k1_ordinal1(A)) & v1_ordinal1(k1_ordinal1(A)) & v2_ordinal1(k1_ordinal1(A)) & v3_ordinal1(k1_ordinal1(A)) ) ) ), file(ordinal1,fc3_ordinal1), []). fof(fc3_ordinal2,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) & v3_ordinal1(B) ) => ( v1_ordinal1(k1_funct_1(A,B)) & v2_ordinal1(k1_funct_1(A,B)) & v3_ordinal1(k1_funct_1(A,B)) ) ) ), file(ordinal2,fc3_ordinal2), []). fof(fc3_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => ( v1_relat_1(k5_zf_lang(A,B)) & v1_funct_1(k5_zf_lang(A,B)) & v1_finset_1(k5_zf_lang(A,B)) & v1_finseq_1(k5_zf_lang(A,B)) & v1_zf_lang(k5_zf_lang(A,B)) ) ) ), file(zf_lang,fc3_zf_lang), []). fof(fc4_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) ) ) ), file(funct_1,fc4_funct_1), []). fof(fc4_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ( v1_ordinal1(k3_tarski(A)) & v2_ordinal1(k3_tarski(A)) & v3_ordinal1(k3_tarski(A)) ) ) ), file(ordinal1,fc4_ordinal1), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc4_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_relat_1(k6_zf_lang(A)) & v1_funct_1(k6_zf_lang(A)) & v1_finset_1(k6_zf_lang(A)) & v1_finseq_1(k6_zf_lang(A)) & v1_zf_lang(k6_zf_lang(A)) ) ) ), file(zf_lang,fc4_zf_lang), []). fof(fc5_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k8_relat_1(A,B)) & v1_funct_1(k8_relat_1(A,B)) ) ) ), file(funct_1,fc5_funct_1), []). fof(fc5_ordinal1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) ) => ( v1_ordinal1(k1_relat_1(A)) & v2_ordinal1(k1_relat_1(A)) & v3_ordinal1(k1_relat_1(A)) ) ) ), file(ordinal1,fc5_ordinal1), []). fof(fc5_zf_lang,axiom,( ! [A,B] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & v1_zf_lang(B) & m1_finseq_1(B,k5_numbers) ) => ( v1_relat_1(k7_zf_lang(A,B)) & v1_funct_1(k7_zf_lang(A,B)) & v1_finset_1(k7_zf_lang(A,B)) & v1_finseq_1(k7_zf_lang(A,B)) & v1_zf_lang(k7_zf_lang(A,B)) ) ) ), file(zf_lang,fc5_zf_lang), []). fof(fc6_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v2_relat_1(A) & v1_funct_1(A) ) => v1_setfam_1(k2_relat_1(A)) ) ), file(funct_1,fc6_funct_1), []). fof(fc6_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & v1_zf_lang(B) & m1_finseq_1(B,k5_numbers) ) => ( v1_relat_1(k8_zf_lang(A,B)) & v1_funct_1(k8_zf_lang(A,B)) & v1_finset_1(k8_zf_lang(A,B)) & v1_finseq_1(k8_zf_lang(A,B)) & v1_zf_lang(k8_zf_lang(A,B)) ) ) ), file(zf_lang,fc6_zf_lang), []). fof(fc7_classes2,axiom,( ! [A] : ( v3_ordinal1(A) => v1_ordinal1(k4_classes1(A)) ) ), file(classes2,fc7_classes2), []). fof(fraenkel_a_0_0_zf_lang,axiom,( ! [A] : ( r2_hidden(A,a_0_0_zf_lang) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = B & r1_xreal_0(5,B) ) ) ), file(zf_lang,a_0_0_zf_lang), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k3_zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_ordinal4(B,A) & m1_ordinal4(C,A) ) => k3_zf_refle(A,B,B) = B ) ), file(zf_refle,k3_zf_refle), []). fof(irreflexivity_r2_xboole_0,axiom,( ! [A,B] : ~ r2_xboole_0(A,A) ), file(xboole_0,r2_xboole_0), []). fof(rc1_card_1,axiom,( ? [A] : v1_card_1(A) ), file(card_1,rc1_card_1), []). fof(rc1_classes2,axiom,( ? [A] : ( v1_ordinal1(A) & ~ v1_xboole_0(A) & v1_classes1(A) & v2_classes1(A) & v1_classes2(A) ) ), file(classes2,rc1_classes2), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_ordinal1,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc1_ordinal1), []). fof(rc1_ordinal2,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal1(A) ) ), file(ordinal2,rc1_ordinal2), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_zf_lang,axiom,( ? [A] : ( m1_finseq_1(A,k5_numbers) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_zf_lang(A) ) ), file(zf_lang,rc1_zf_lang), []). fof(rc1_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ? [B] : ( m1_subset_1(B,A) & ~ v1_xboole_0(B) ) ) ), file(zf_refle,rc1_zf_refle), []). fof(rc2_card_1,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v1_finset_1(A) & v1_card_1(A) ) ), file(card_1,rc2_card_1), []). fof(rc2_classes2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ? [B] : ( m1_subset_1(B,A) & ~ v1_xboole_0(B) ) ) ), file(classes2,rc2_classes2), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_ordinal1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc2_ordinal1), []). fof(rc2_ordinal2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) ), file(ordinal2,rc2_ordinal2), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ? [B] : ( m1_ordinal1(B,A) & v1_relat_1(B) & v2_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_zf_refle(B,A) ) ) ), file(zf_refle,rc2_zf_refle), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_ordinal1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc3_ordinal1), []). fof(rc4_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). fof(rc4_ordinal1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) ) ), file(ordinal1,rc4_ordinal1), []). fof(rc5_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v2_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc5_funct_1), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => k1_fraenkel(A,B) = k1_funct_2(A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(redefinition_k2_ordinal1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v3_ordinal1(B) ) => k2_ordinal1(A,B) = k7_relat_1(A,B) ) ), file(ordinal1,k2_ordinal1), []). fof(redefinition_k2_zf_model,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => k2_zf_model(A) = k1_zf_model(A) ) ), file(zf_model,k2_zf_model), []). fof(redefinition_k2_zf_refle,axiom,( ! [A,B,C] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) & ~ v1_xboole_0(B) & v1_classes2(B) & m1_ordinal4(C,B) ) => k2_zf_refle(A,B,C) = k1_zf_refle(A,B,C) ) ), file(zf_refle,k2_zf_refle), []). fof(redefinition_k3_zf_lang,axiom,( ! [A] : ( m1_subset_1(A,k1_zf_lang) => k3_zf_lang(A) = k5_finseq_1(A) ) ), file(zf_lang,k3_zf_lang), []). fof(redefinition_k3_zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_ordinal4(B,A) & m1_ordinal4(C,A) ) => k3_zf_refle(A,B,C) = k2_xboole_0(B,C) ) ), file(zf_refle,k3_zf_refle), []). fof(redefinition_k4_ordinal4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) & m1_ordinal4(C,A) ) => k4_ordinal4(A,B,C) = k1_funct_1(B,C) ) ), file(ordinal4,k4_ordinal4), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k4_zf_refle,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) ) => k4_zf_refle(A,B) = k3_card_3(B) ) ), file(zf_refle,k4_zf_refle), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_ordinal4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) & m2_ordinal4(C,A) ) => k5_ordinal4(A,B,C) = k5_relat_1(B,C) ) ), file(ordinal4,k5_ordinal4), []). fof(redefinition_k5_zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) & m1_ordinal4(C,A) ) => k5_zf_refle(A,B,C) = k1_funct_1(B,C) ) ), file(zf_refle,k5_zf_refle), []). fof(redefinition_k6_ordinal4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_ordinal4(B,A) ) => k6_ordinal4(A,B) = k1_ordinal1(B) ) ), file(ordinal4,k6_ordinal4), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ! [D] : ( m2_fraenkel(D,A,B,C) <=> m1_subset_1(D,C) ) ) ), file(fraenkel,m2_fraenkel), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), []). fof(redefinition_r1_ordinal1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => ( r1_ordinal1(A,B) <=> r1_tarski(A,B) ) ) ), file(ordinal1,r1_ordinal1), []). fof(redefinition_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) <=> r2_tarski(A,B) ) ), file(wellord2,r2_wellord2), []). fof(reflexivity_r1_ordinal1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => r1_ordinal1(A,A) ) ), file(ordinal1,r1_ordinal1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(reflexivity_r2_wellord2,axiom,( ! [A,B] : r2_wellord2(A,A) ), file(wellord2,r2_wellord2), []). fof(s1_partfun1__e18_36_4_6__zf_refle,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) & v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) & v1_funct_1(D) & v1_funct_2(D,k1_zf_lang,k4_zf_refle(A,B)) & m2_relset_1(D,k1_zf_lang,k4_zf_refle(A,B)) & m1_ordinal4(E,A) & m1_subset_1(F,k5_zf_refle(A,B,E)) ) => ? [G] : ( v1_relat_1(G) & v1_funct_1(G) & k1_relat_1(G) = k1_zf_lang & ! [H] : ( r2_hidden(H,k1_zf_lang) => ( ( r2_hidden(H,k2_zf_model(k6_zf_lang(k24_zf_lang(C)))) => k1_funct_1(G,H) = k1_funct_1(D,H) ) & ( ~ r2_hidden(H,k2_zf_model(k6_zf_lang(k24_zf_lang(C)))) => k1_funct_1(G,H) = F ) ) ) ) ) ), file(zf_refle,s1_partfun1__e18_36_4_6__zf_refle), []). fof(s1_zf_lang__e8_36__zf_refle,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) ) => ( ( ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( v7_zf_lang(C) => ? [D] : ( m2_ordinal4(D,A) & v2_ordinal2(D) & v3_ordinal2(D) & ! [E] : ( m1_ordinal4(E,A) => ( k4_ordinal4(A,D,E) = E => ( k1_xboole_0 = E | ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k1_zf_lang,k5_zf_refle(A,B,E)) & m2_relset_1(F,k1_zf_lang,k5_zf_refle(A,B,E)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,E),k4_zf_refle(A,B),F),C) <=> r1_zf_model(k5_zf_refle(A,B,E),F,C) ) ) ) ) ) ) ) ) & ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( ( v4_zf_lang(C) & ? [G] : ( m2_ordinal4(G,A) & v2_ordinal2(G) & v3_ordinal2(G) & ! [H] : ( m1_ordinal4(H,A) => ( k4_ordinal4(A,G,H) = H => ( k1_xboole_0 = H | ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,k1_zf_lang,k5_zf_refle(A,B,H)) & m2_relset_1(I,k1_zf_lang,k5_zf_refle(A,B,H)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,H),k4_zf_refle(A,B),I),k20_zf_lang(C)) <=> r1_zf_model(k5_zf_refle(A,B,H),I,k20_zf_lang(C)) ) ) ) ) ) ) ) => ? [J] : ( m2_ordinal4(J,A) & v2_ordinal2(J) & v3_ordinal2(J) & ! [K] : ( m1_ordinal4(K,A) => ( k4_ordinal4(A,J,K) = K => ( k1_xboole_0 = K | ! [L] : ( ( v1_funct_1(L) & v1_funct_2(L,k1_zf_lang,k5_zf_refle(A,B,K)) & m2_relset_1(L,k1_zf_lang,k5_zf_refle(A,B,K)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,K),k4_zf_refle(A,B),L),C) <=> r1_zf_model(k5_zf_refle(A,B,K),L,C) ) ) ) ) ) ) ) ) & ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( ( v5_zf_lang(C) & ? [M] : ( m2_ordinal4(M,A) & v2_ordinal2(M) & v3_ordinal2(M) & ! [N] : ( m1_ordinal4(N,A) => ( k4_ordinal4(A,M,N) = N => ( k1_xboole_0 = N | ! [O] : ( ( v1_funct_1(O) & v1_funct_2(O,k1_zf_lang,k5_zf_refle(A,B,N)) & m2_relset_1(O,k1_zf_lang,k5_zf_refle(A,B,N)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,N),k4_zf_refle(A,B),O),k21_zf_lang(C)) <=> r1_zf_model(k5_zf_refle(A,B,N),O,k21_zf_lang(C)) ) ) ) ) ) ) & ? [P] : ( m2_ordinal4(P,A) & v2_ordinal2(P) & v3_ordinal2(P) & ! [Q] : ( m1_ordinal4(Q,A) => ( k4_ordinal4(A,P,Q) = Q => ( k1_xboole_0 = Q | ! [R] : ( ( v1_funct_1(R) & v1_funct_2(R,k1_zf_lang,k5_zf_refle(A,B,Q)) & m2_relset_1(R,k1_zf_lang,k5_zf_refle(A,B,Q)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,Q),k4_zf_refle(A,B),R),k22_zf_lang(C)) <=> r1_zf_model(k5_zf_refle(A,B,Q),R,k22_zf_lang(C)) ) ) ) ) ) ) ) => ? [S] : ( m2_ordinal4(S,A) & v2_ordinal2(S) & v3_ordinal2(S) & ! [T] : ( m1_ordinal4(T,A) => ( k4_ordinal4(A,S,T) = T => ( k1_xboole_0 = T | ! [U] : ( ( v1_funct_1(U) & v1_funct_2(U,k1_zf_lang,k5_zf_refle(A,B,T)) & m2_relset_1(U,k1_zf_lang,k5_zf_refle(A,B,T)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,T),k4_zf_refle(A,B),U),C) <=> r1_zf_model(k5_zf_refle(A,B,T),U,C) ) ) ) ) ) ) ) ) & ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( ( v6_zf_lang(C) & ? [V] : ( m2_ordinal4(V,A) & v2_ordinal2(V) & v3_ordinal2(V) & ! [W] : ( m1_ordinal4(W,A) => ( k4_ordinal4(A,V,W) = W => ( k1_xboole_0 = W | ! [X] : ( ( v1_funct_1(X) & v1_funct_2(X,k1_zf_lang,k5_zf_refle(A,B,W)) & m2_relset_1(X,k1_zf_lang,k5_zf_refle(A,B,W)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,W),k4_zf_refle(A,B),X),k24_zf_lang(C)) <=> r1_zf_model(k5_zf_refle(A,B,W),X,k24_zf_lang(C)) ) ) ) ) ) ) ) => ? [Y] : ( m2_ordinal4(Y,A) & v2_ordinal2(Y) & v3_ordinal2(Y) & ! [Z] : ( m1_ordinal4(Z,A) => ( k4_ordinal4(A,Y,Z) = Z => ( k1_xboole_0 = Z | ! [A1] : ( ( v1_funct_1(A1) & v1_funct_2(A1,k1_zf_lang,k5_zf_refle(A,B,Z)) & m2_relset_1(A1,k1_zf_lang,k5_zf_refle(A,B,Z)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,Z),k4_zf_refle(A,B),A1),C) <=> r1_zf_model(k5_zf_refle(A,B,Z),A1,C) ) ) ) ) ) ) ) ) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ? [B1] : ( m2_ordinal4(B1,A) & v2_ordinal2(B1) & v3_ordinal2(B1) & ! [C1] : ( m1_ordinal4(C1,A) => ( k4_ordinal4(A,B1,C1) = C1 => ( k1_xboole_0 = C1 | ! [D1] : ( ( v1_funct_1(D1) & v1_funct_2(D1,k1_zf_lang,k5_zf_refle(A,B,C1)) & m2_relset_1(D1,k1_zf_lang,k5_zf_refle(A,B,C1)) ) => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,C1),k4_zf_refle(A,B),D1),C) <=> r1_zf_model(k5_zf_refle(A,B,C1),D1,C) ) ) ) ) ) ) ) ) ) ), file(zf_refle,s1_zf_lang__e8_36__zf_refle), []). fof(s2_zf_refle__e8_36_4__zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) & v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( ! [D] : ( m1_subset_1(D,k1_fraenkel(k1_zf_lang,k4_zf_refle(A,B))) => ? [E] : ( m1_ordinal4(E,A) & ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k1_zf_lang,k4_zf_refle(A,B)) & m2_relset_1(F,k1_zf_lang,k4_zf_refle(A,B)) & D = F & ~ ( ! [G] : ( m2_subset_1(G,A,k4_zf_refle(A,B)) => ~ ( r2_hidden(G,k1_funct_1(B,E)) & r1_zf_model(k4_zf_refle(A,B),k1_zf_lang1(k4_zf_refle(A,B),F,k23_zf_lang(C),G),k6_zf_lang(k24_zf_lang(C))) ) ) & ~ ( E = k2_ordinal4(A) & ! [G] : ( m2_subset_1(G,A,k4_zf_refle(A,B)) => ~ r1_zf_model(k4_zf_refle(A,B),k1_zf_lang1(k4_zf_refle(A,B),F,k23_zf_lang(C),G),k6_zf_lang(k24_zf_lang(C))) ) ) ) ) ) ) => ? [D] : ( v1_relat_1(D) & v1_funct_1(D) & k1_relat_1(D) = k1_fraenkel(k1_zf_lang,k4_zf_refle(A,B)) & ! [E] : ( m1_subset_1(E,k1_fraenkel(k1_zf_lang,k4_zf_refle(A,B))) => ? [H] : ( m1_ordinal4(H,A) & H = k1_funct_1(D,E) & ? [I] : ( v1_funct_1(I) & v1_funct_2(I,k1_zf_lang,k4_zf_refle(A,B)) & m2_relset_1(I,k1_zf_lang,k4_zf_refle(A,B)) & E = I & ~ ( ! [J] : ( m2_subset_1(J,A,k4_zf_refle(A,B)) => ~ ( r2_hidden(J,k1_funct_1(B,H)) & r1_zf_model(k4_zf_refle(A,B),k1_zf_lang1(k4_zf_refle(A,B),I,k23_zf_lang(C),J),k6_zf_lang(k24_zf_lang(C))) ) ) & ~ ( H = k2_ordinal4(A) & ! [J] : ( m2_subset_1(J,A,k4_zf_refle(A,B)) => ~ r1_zf_model(k4_zf_refle(A,B),k1_zf_lang1(k4_zf_refle(A,B),I,k23_zf_lang(C),J),k6_zf_lang(k24_zf_lang(C))) ) ) ) ) & ! [K] : ( m1_ordinal4(K,A) => ( ? [L] : ( v1_funct_1(L) & v1_funct_2(L,k1_zf_lang,k4_zf_refle(A,B)) & m2_relset_1(L,k1_zf_lang,k4_zf_refle(A,B)) & E = L & ~ ( ! [M] : ( m2_subset_1(M,A,k4_zf_refle(A,B)) => ~ ( r2_hidden(M,k1_funct_1(B,K)) & r1_zf_model(k4_zf_refle(A,B),k1_zf_lang1(k4_zf_refle(A,B),L,k23_zf_lang(C),M),k6_zf_lang(k24_zf_lang(C))) ) ) & ~ ( K = k2_ordinal4(A) & ! [M] : ( m2_subset_1(M,A,k4_zf_refle(A,B)) => ~ r1_zf_model(k4_zf_refle(A,B),k1_zf_lang1(k4_zf_refle(A,B),L,k23_zf_lang(C),M),k6_zf_lang(k24_zf_lang(C))) ) ) ) ) => r1_tarski(H,K) ) ) ) ) ) ) ) ), file(zf_refle,s2_zf_refle__e8_36_4__zf_refle), []). fof(s3_card_3__e5_36_4_6_1__zf_refle,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) & v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) & v1_funct_1(D) & v1_funct_2(D,k1_zf_lang,k4_zf_refle(A,B)) & m2_relset_1(D,k1_zf_lang,k4_zf_refle(A,B)) ) => ( ( k2_zf_model(k6_zf_lang(k24_zf_lang(C))) != k1_xboole_0 & ! [E,F] : ( ! [G] : ( m1_ordinal4(G,A) => ( r2_hidden(k1_funct_1(D,E),k5_zf_refle(A,B,G)) => r2_hidden(k1_funct_1(D,F),k5_zf_refle(A,B,G)) ) ) | ! [H] : ( m1_ordinal4(H,A) => ( r2_hidden(k1_funct_1(D,F),k5_zf_refle(A,B,H)) => r2_hidden(k1_funct_1(D,E),k5_zf_refle(A,B,H)) ) ) ) & ! [E,F,I] : ( ( ! [J] : ( m1_ordinal4(J,A) => ( r2_hidden(k1_funct_1(D,E),k5_zf_refle(A,B,J)) => r2_hidden(k1_funct_1(D,F),k5_zf_refle(A,B,J)) ) ) & ! [K] : ( m1_ordinal4(K,A) => ( r2_hidden(k1_funct_1(D,F),k5_zf_refle(A,B,K)) => r2_hidden(k1_funct_1(D,I),k5_zf_refle(A,B,K)) ) ) ) => ! [L] : ( m1_ordinal4(L,A) => ( r2_hidden(k1_funct_1(D,E),k5_zf_refle(A,B,L)) => r2_hidden(k1_funct_1(D,I),k5_zf_refle(A,B,L)) ) ) ) ) => ? [E] : ( r2_hidden(E,k2_zf_model(k6_zf_lang(k24_zf_lang(C)))) & ! [F] : ( r2_hidden(F,k2_zf_model(k6_zf_lang(k24_zf_lang(C)))) => ! [M] : ( m1_ordinal4(M,A) => ( r2_hidden(k1_funct_1(D,E),k5_zf_refle(A,B,M)) => r2_hidden(k1_funct_1(D,F),k5_zf_refle(A,B,M)) ) ) ) ) ) ) ), file(zf_refle,s3_card_3__e5_36_4_6_1__zf_refle), []). fof(s3_zf_refle__e13_36_4__zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) & v1_relat_1(C) & v1_funct_1(C) ) => ( ( ! [D] : ( m1_ordinal4(D,A) => ! [E] : ( m1_ordinal4(E,A) => ! [F] : ( m1_ordinal4(F,A) => ( ( E = k7_ordinal2(k9_relat_1(C,k1_fraenkel(k1_zf_lang,k5_zf_refle(A,B,D)))) & F = k7_ordinal2(k9_relat_1(C,k1_fraenkel(k1_zf_lang,k5_zf_refle(A,B,D)))) ) => E = F ) ) ) ) & ! [D] : ( m1_ordinal4(D,A) => ? [E] : ( m1_ordinal4(E,A) & E = k7_ordinal2(k9_relat_1(C,k1_fraenkel(k1_zf_lang,k5_zf_refle(A,B,D)))) ) ) ) => ? [D] : ( m2_ordinal4(D,A) & ! [E] : ( m1_ordinal4(E,A) => k4_ordinal4(A,D,E) = k7_ordinal2(k9_relat_1(C,k1_fraenkel(k1_zf_lang,k5_zf_refle(A,B,E)))) ) ) ) ) ), file(zf_refle,s3_zf_refle__e13_36_4__zf_refle), []). fof(s4_zf_refle__e15_36_4__zf_refle,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) ) => ? [C] : ( m2_ordinal4(C,A) & k4_ordinal4(A,C,k2_ordinal4(A)) = k4_ordinal4(A,B,k2_ordinal4(A)) & ! [D] : ( m1_ordinal4(D,A) => k4_ordinal4(A,C,k6_ordinal4(A,D)) = k6_ordinal4(A,k3_zf_refle(A,k4_ordinal4(A,B,k6_ordinal4(A,D)),k4_ordinal4(A,C,D))) ) & ! [D] : ( m1_ordinal4(D,A) => ( v4_ordinal1(D) => ( D = k2_ordinal4(A) | k4_ordinal4(A,C,D) = k2_zf_refle(k2_ordinal1(C,D),A,D) ) ) ) ) ) ), file(zf_refle,s4_zf_refle__e15_36_4__zf_refle), []). fof(s5_zf_refle__e26_36_4__zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) & m2_ordinal4(C,A) ) => ( ( r1_tarski(k1_funct_1(B,k2_ordinal4(A)),k1_funct_1(C,k2_ordinal4(A))) & ! [D] : ( m1_ordinal4(D,A) => ( r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) => r1_tarski(k1_funct_1(B,k6_ordinal4(A,D)),k1_funct_1(C,k6_ordinal4(A,D))) ) ) & ! [D] : ( m1_ordinal4(D,A) => ( ( v4_ordinal1(D) & ! [E] : ( m1_ordinal4(E,A) => ( r2_hidden(E,D) => r1_tarski(k1_funct_1(B,E),k1_funct_1(C,E)) ) ) ) => ( D = k2_ordinal4(A) | r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) => ! [D] : ( m1_ordinal4(D,A) => r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ), file(zf_refle,s5_zf_refle__e26_36_4__zf_refle), []). fof(s5_zf_refle__e7_36_4_4__zf_refle,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m2_ordinal4(B,A) & m1_ordinal4(C,A) ) => ( ( ( r2_hidden(C,k2_ordinal4(A)) => r2_hidden(k4_ordinal4(A,B,C),k4_ordinal4(A,B,k2_ordinal4(A))) ) & ! [D] : ( m1_ordinal4(D,A) => ( ( r2_hidden(C,D) => r2_hidden(k4_ordinal4(A,B,C),k4_ordinal4(A,B,D)) ) => ( r2_hidden(C,k6_ordinal4(A,D)) => r2_hidden(k4_ordinal4(A,B,C),k4_ordinal4(A,B,k6_ordinal4(A,D))) ) ) ) & ! [D] : ( m1_ordinal4(D,A) => ( ( v4_ordinal1(D) & ! [E] : ( m1_ordinal4(E,A) => ( r2_hidden(E,D) => ( r2_hidden(C,E) => r2_hidden(k4_ordinal4(A,B,C),k4_ordinal4(A,B,E)) ) ) ) ) => ( D = k2_ordinal4(A) | ( r2_hidden(C,D) => r2_hidden(k4_ordinal4(A,B,C),k4_ordinal4(A,B,D)) ) ) ) ) ) => ! [D] : ( m1_ordinal4(D,A) => ( r2_hidden(C,D) => r2_hidden(k4_ordinal4(A,B,C),k4_ordinal4(A,B,D)) ) ) ) ) ), file(zf_refle,s5_zf_refle__e7_36_4_4__zf_refle), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(spc3_boole,axiom,( ~ v1_xboole_0(3) ), file(boole,spc3_boole), []). fof(spc3_numerals,axiom, ( v2_xreal_0(3) & m1_subset_1(3,k5_numbers) ), file(numerals,spc3_numerals), []). fof(spc4_boole,axiom,( ~ v1_xboole_0(4) ), file(boole,spc4_boole), []). fof(spc4_numerals,axiom, ( v2_xreal_0(4) & m1_subset_1(4,k5_numbers) ), file(numerals,spc4_numerals), []). fof(spc5_boole,axiom,( ~ v1_xboole_0(5) ), file(boole,spc5_boole), []). fof(spc5_numerals,axiom, ( v2_xreal_0(5) & m1_subset_1(5,k5_numbers) ), file(numerals,spc5_numerals), []). fof(symmetry_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) => r2_wellord2(B,A) ) ), file(wellord2,r2_wellord2), []). fof(t10_ordinal1,axiom,( ! [A] : r2_hidden(A,k1_ordinal1(A)) ), file(ordinal1,t10_ordinal1), []). fof(t10_ordinal3,axiom,( ! [A] : ( v3_ordinal1(A) => ( A != k1_xboole_0 => r2_hidden(k1_xboole_0,A) ) ) ), file(ordinal3,t10_ordinal3), []). fof(t10_ordinal4,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ! [B] : ( v3_ordinal1(B) => ( ( v2_ordinal2(A) & r2_hidden(B,k1_relat_1(A)) ) => r1_ordinal1(B,k1_funct_1(A,B)) ) ) ) ), file(ordinal4,t10_ordinal4), []). fof(t10_zfmodel1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k1_zf_lang,C) & m2_relset_1(D,k1_zf_lang,C) ) => ( r1_zf_model(C,D,B) => ( r2_hidden(A,k2_zf_model(B)) | r1_zf_model(C,D,k8_zf_lang(A,B)) ) ) ) ) ) ) ), file(zfmodel1,t10_zfmodel1), []). fof(t11_funct_2,axiom,( ! [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), []). fof(t11_relset_1,axiom,( ! [A,B,C] : ( v1_relat_1(C) => ( ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) => m2_relset_1(C,A,B) ) ) ), file(relset_1,t11_relset_1), []). fof(t12_zf_model,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m2_relset_1(B,k1_zf_lang,A) ) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ! [D] : ( m2_subset_1(D,k5_numbers,k1_zf_lang) => ( r1_zf_model(A,B,k4_zf_lang(C,D)) <=> k8_funct_2(k1_zf_lang,A,B,C) = k8_funct_2(k1_zf_lang,A,B,D) ) ) ) ) ) ), file(zf_model,t12_zf_model), []). fof(t13_ordinal4,axiom,( ! [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) ) => ~ ( v2_ordinal2(A) & v2_ordinal2(B) & ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) ) => ~ ( C = k5_relat_1(B,A) & v2_ordinal2(C) ) ) ) ) ) ), file(ordinal4,t13_ordinal4), []). fof(t13_zf_model,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m2_relset_1(B,k1_zf_lang,A) ) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ! [D] : ( m2_subset_1(D,k5_numbers,k1_zf_lang) => ( r1_zf_model(A,B,k5_zf_lang(C,D)) <=> r2_hidden(k8_funct_2(k1_zf_lang,A,B,C),k8_funct_2(k1_zf_lang,A,B,D)) ) ) ) ) ) ), file(zf_model,t13_zf_model), []). fof(t14_zf_model,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m2_relset_1(B,k1_zf_lang,A) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( r1_zf_model(A,B,C) <=> ~ r1_zf_model(A,B,k6_zf_lang(C)) ) ) ) ) ), file(zf_model,t14_zf_model), []). fof(t15_ordinal4,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ! [B] : ( v3_ordinal1(B) => ( v2_ordinal2(A) => v2_ordinal2(k2_ordinal1(A,B)) ) ) ) ), file(ordinal4,t15_ordinal4), []). fof(t15_zf_model,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m2_relset_1(B,k1_zf_lang,A) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ! [D] : ( ( v1_zf_lang(D) & m2_finseq_1(D,k5_numbers) ) => ( r1_zf_model(A,B,k7_zf_lang(C,D)) <=> ( r1_zf_model(A,B,C) & r1_zf_model(A,B,D) ) ) ) ) ) ) ), file(zf_model,t15_zf_model), []). fof(t16_zf_model,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k1_zf_lang,A) & m2_relset_1(B,k1_zf_lang,A) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ! [D] : ( m2_subset_1(D,k5_numbers,k1_zf_lang) => ( r1_zf_model(A,B,k8_zf_lang(D,C)) <=> ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,A) & m2_relset_1(E,k1_zf_lang,A) ) => ( ! [F] : ( m2_subset_1(F,k5_numbers,k1_zf_lang) => ( k8_funct_2(k1_zf_lang,A,E,F) != k8_funct_2(k1_zf_lang,A,B,F) => D = F ) ) => r1_zf_model(A,E,C) ) ) ) ) ) ) ) ), file(zf_model,t16_zf_model), []). fof(t17_ordinal4,axiom,( ! [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) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) & v1_ordinal2(C) ) => ( ( v2_ordinal2(A) & v3_ordinal2(A) & v3_ordinal2(B) & C = k5_relat_1(A,B) ) => v3_ordinal2(C) ) ) ) ) ), file(ordinal4,t17_ordinal4), []). fof(t17_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal4(B,A) => ! [C] : ( m2_ordinal4(C,A) => ( k2_zf_refle(C,A,B) = k3_card_3(k2_ordinal1(C,B)) & k2_zf_refle(k2_ordinal1(C,B),A,B) = k3_card_3(k2_ordinal1(C,B)) ) ) ) ) ), file(zf_refle,t17_zf_refle), []). fof(t19_ordinal1,axiom,( ! [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), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_classes2,axiom,( ! [A,B] : ( ( v1_classes1(A) & r2_hidden(B,A) ) => ( ~ r2_wellord2(B,A) & r2_hidden(k1_card_1(B),k1_card_1(A)) ) ) ), file(classes2,t1_classes2), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), []). fof(t21_card_1,axiom,( ! [A,B] : ( r2_wellord2(A,B) <=> k1_card_1(A) = k1_card_1(B) ) ), file(card_1,t21_card_1), []). fof(t21_ordinal1,axiom,( ! [A] : ( v1_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r2_xboole_0(A,B) => r2_hidden(A,B) ) ) ) ), file(ordinal1,t21_ordinal1), []). fof(t22_funct_1,axiom,( ! [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), []). fof(t22_ordinal1,axiom,( ! [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), []). fof(t23_ordinal1,axiom,( ! [A,B] : ( v3_ordinal1(B) => ( r2_hidden(A,B) => v3_ordinal1(A) ) ) ), file(ordinal1,t23_ordinal1), []). fof(t24_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ~ ( ~ r2_hidden(A,B) & A != B & ~ r2_hidden(B,A) ) ) ) ), file(ordinal1,t24_ordinal1), []). fof(t24_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal4(B,A) => ! [C] : ( ( v2_relat_1(C) & v1_zf_refle(C,A) & m1_ordinal1(C,A) ) => r1_tarski(k5_zf_refle(A,C,B),k4_zf_refle(A,C)) ) ) ) ), file(zf_refle,t24_zf_refle), []). fof(t25_zf_refle,axiom,( r2_tarski(k5_numbers,k1_zf_lang) ), file(zf_refle,t25_zf_refle), []). fof(t26_ordinal2,axiom,( ! [A] : ( v3_ordinal1(A) => k7_ordinal2(A) = A ) ), file(ordinal2,t26_ordinal2), []). fof(t27_ordinal2,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r2_hidden(A,B) => r2_hidden(A,k7_ordinal2(B)) ) ) ), file(ordinal2,t27_ordinal2), []). fof(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t28_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( r2_hidden(B,A) => r2_hidden(k7_ordinal2(B),A) ) ) ), file(zf_refle,t28_zf_refle), []). fof(t29_ordinal2,axiom,( ! [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), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_classes1,axiom,( ! [A] : ( v2_classes1(A) <=> ( v1_classes1(A) & ! [B] : ( r2_hidden(B,A) => r2_hidden(k1_zfmisc_1(B),A) ) & ! [B] : ( ( r1_tarski(B,A) & r2_hidden(k1_card_1(B),k1_card_1(A)) ) => r2_hidden(B,A) ) ) ) ), file(classes1,t2_classes1), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t34_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( v3_ordinal1(B) => ( r2_hidden(A,k1_ordinal1(B)) <=> r1_ordinal1(A,B) ) ) ) ), file(ordinal1,t34_ordinal1), []). fof(t35_funct_1,axiom,( ! [A,B] : ( r2_hidden(B,A) => k1_funct_1(k6_relat_1(A),B) = B ) ), file(funct_1,t35_funct_1), []). fof(t35_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ( k2_ordinal4(A) = k1_xboole_0 & k3_ordinal4(A) = k4_ordinal2 ) ) ), file(ordinal4,t35_ordinal4), []). fof(t36_ordinal4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal4(B,A) => ! [C] : ( m2_ordinal4(C,A) => r2_hidden(B,k1_relat_1(C)) ) ) ) ), file(ordinal4,t36_ordinal4), []). fof(t3_card_2,axiom,( ! [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), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t41_ordinal1,axiom,( ! [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), []). fof(t43_funct_1,axiom,( ! [A,B] : k5_relat_1(k6_relat_1(B),k6_relat_1(A)) = k6_relat_1(k3_xboole_0(A,B)) ), file(funct_1,t43_funct_1), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t53_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( v2_zf_lang(A) => A = k4_zf_lang(k18_zf_lang(A),k19_zf_lang(A)) ) ) ), file(zf_lang,t53_zf_lang), []). fof(t54_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( v3_zf_lang(A) => A = k5_zf_lang(k18_zf_lang(A),k19_zf_lang(A)) ) ) ), file(zf_lang,t54_zf_lang), []). fof(t58_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( v5_zf_lang(A) => A = k7_zf_lang(k21_zf_lang(A),k22_zf_lang(A)) ) ) ), file(zf_lang,t58_zf_lang), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t62_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( v6_zf_lang(A) => A = k8_zf_lang(k23_zf_lang(A),k24_zf_lang(A)) ) ) ), file(zf_lang,t62_zf_lang), []). fof(t63_funct_5,axiom,( ! [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), []). fof(t67_classes2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(C) & v1_classes2(C) ) => ( ( r2_hidden(A,C) & r2_hidden(B,C) ) => ( r2_hidden(k2_zfmisc_1(A,B),C) & r2_hidden(k1_funct_2(A,B),C) ) ) ) ), file(classes2,t67_classes2), []). fof(t68_funct_5,axiom,( ! [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), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t70_funct_1,axiom,( ! [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), []). fof(t71_relat_1,axiom,( ! [A] : ( k1_relat_1(k6_relat_1(A)) = A & k2_relat_1(k6_relat_1(A)) = A ) ), file(relat_1,t71_relat_1), []). fof(t72_funct_1,axiom,( ! [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), []). fof(t73_funct_1,axiom,( ! [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), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_ordinal1,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & r1_tarski(B,A) ) ), file(ordinal1,t7_ordinal1), []). fof(t7_xboole_1,axiom,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), []). fof(t80_zf_lang1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k1_zf_lang,C) & m2_relset_1(D,k1_zf_lang,C) ) => ( r1_zf_model(C,D,k8_zf_lang(B,A)) <=> ! [E] : ( m1_subset_1(E,C) => r1_zf_model(C,k1_zf_lang1(C,D,B,E),A) ) ) ) ) ) ) ), file(zf_lang1,t80_zf_lang1), []). fof(t84_zf_lang1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,B) & m2_relset_1(C,k1_zf_lang,B) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k1_zf_lang,B) & m2_relset_1(D,k1_zf_lang,B) ) => ( ( ! [E] : ( m2_subset_1(E,k5_numbers,k1_zf_lang) => ( r2_hidden(E,k2_zf_model(A)) => k8_funct_2(k1_zf_lang,B,D,E) = k8_funct_2(k1_zf_lang,B,C,E) ) ) & r1_zf_model(B,C,A) ) => r1_zf_model(B,D,A) ) ) ) ) ) ), file(zf_lang1,t84_zf_lang1), []). fof(t87_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k1_relat_1(k7_relat_1(B,A)),A) ) ), file(relat_1,t87_relat_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_ordinal4,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) => ( ( v4_ordinal1(k1_relat_1(A)) & v2_ordinal2(A) ) => ( k1_relat_1(A) = k1_xboole_0 | ( r1_ordinal2(k8_ordinal2(A),A) & k12_ordinal2(A) = k8_ordinal2(A) ) ) ) ) ), file(ordinal4,t8_ordinal4), []). fof(t8_zf_refle,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal4(B,A) <=> r2_hidden(B,k2_ordinal2(A)) ) ) ), file(zf_refle,t8_zf_refle), []). fof(t90_relat_1,axiom,( ! [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), []). fof(t91_relat_1,axiom,( ! [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), []). fof(t94_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => k7_relat_1(B,A) = k5_relat_1(k6_relat_1(A),B) ) ), file(relat_1,t94_relat_1), []).