% Mizar problem: t3_zf_fund2,zf_fund2,398,45 fof(t3_zf_fund2,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] : ( m1_ordinal4(C,A) => ( r2_hidden(k5_zf_refle(A,B,C),k4_zf_refle(A,B)) & v1_ordinal1(k5_zf_refle(A,B,C)) ) ) & v1_zf_fund2(k4_zf_refle(A,B)) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => ( r1_xboole_0(k1_enumset1(k2_zf_lang(0),k2_zf_lang(1),k2_zf_lang(2)),k6_zf_fund1(C)) => r2_zf_model(k4_zf_refle(A,B),k11_zf_model(C)) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_card_1,cc1_classes2,cc1_finset_1,cc1_finsub_1,cc1_funct_1,cc1_ordinal1,cc1_ordinal2,cc1_relset_1,cc2_card_1,cc2_classes2,cc2_finset_1,cc2_finsub_1,cc2_funct_1,cc2_ordinal1,cc3_card_1,cc3_classes2,cc3_ordinal1,commutativity_k1_nat_1,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_zf_refle,commutativity_k4_subset_1,connectedness_r1_xreal_0,d10_xboole_0,d12_funct_1,d1_enumset1,d1_funct_2,d1_tarski,d1_zf_fund2,d1_zf_lang1,d2_ordinal1,d2_ordinal4,d2_xboole_0,d2_zf_fund2,d2_zf_lang1,d3_tarski,d4_xboole_0,d5_funct_1,d5_zf_refle,dt_k11_zf_model,dt_k12_zf_lang,dt_k13_zf_lang,dt_k1_card_1,dt_k1_enumset1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zf_fund2,dt_k1_zf_lang,dt_k1_zf_lang1,dt_k1_zf_model,dt_k1_zfmisc_1,dt_k1_zfmodel1,dt_k2_funct_2,dt_k2_ordinal1,dt_k2_ordinal2,dt_k2_relat_1,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zf_lang,dt_k2_zf_lang1,dt_k2_zf_model,dt_k2_zfmisc_1,dt_k3_card_1,dt_k3_card_3,dt_k3_zf_lang1,dt_k3_zf_refle,dt_k4_ordinal4,dt_k4_relset_1,dt_k4_subset_1,dt_k4_xboole_0,dt_k4_zf_lang,dt_k4_zf_lang1,dt_k4_zf_refle,dt_k5_numbers,dt_k5_ordinal2,dt_k5_zf_lang,dt_k5_zf_lang1,dt_k5_zf_refle,dt_k6_subset_1,dt_k6_zf_fund1,dt_k6_zf_lang1,dt_k7_ordinal2,dt_k7_relat_1,dt_k7_zf_lang,dt_k8_funct_2,dt_k8_zf_fund1,dt_k8_zf_lang,dt_k9_relat_1,dt_m1_finseq_1,dt_m1_ordinal1,dt_m1_ordinal4,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_ordinal4,dt_m2_relset_1,dt_m2_subset_1,existence_m1_finseq_1,existence_m1_ordinal1,existence_m1_ordinal4,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_ordinal4,existence_m2_relset_1,existence_m2_subset_1,fc12_finset_1,fc13_finset_1,fc14_finset_1,fc1_card_1,fc1_finset_1,fc1_finsub_1,fc1_ordinal2,fc1_zf_lang,fc2_card_1,fc2_classes2,fc2_ordinal1,fc2_ordinal2,fc2_zf_lang,fc3_finset_1,fc3_ordinal2,fc3_zf_lang,fc4_funct_1,fc5_ordinal1,fc5_zf_lang,fc6_funct_1,fc6_zf_lang,fc9_finset_1,fraenkel_a_3_0_zf_fund2,fraenkel_a_7_0_zf_fund2,idempotence_k2_xboole_0,idempotence_k3_zf_refle,idempotence_k4_subset_1,l4_zf_fund2,l6_zf_fund2,l7_zf_fund2,l8_zf_fund2,l9_zf_fund2,rc1_card_1,rc1_classes2,rc1_finset_1,rc1_funct_1,rc1_ordinal1,rc1_ordinal2,rc1_zf_lang,rc2_card_1,rc2_classes2,rc2_finset_1,rc2_funct_1,rc2_ordinal1,rc2_ordinal2,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc4_finset_1,rc4_funct_1,rc4_ordinal1,rc5_funct_1,redefinition_k1_nat_1,redefinition_k2_funct_2,redefinition_k2_ordinal1,redefinition_k2_zf_model,redefinition_k3_zf_refle,redefinition_k4_ordinal4,redefinition_k4_relset_1,redefinition_k4_subset_1,redefinition_k4_zf_lang1,redefinition_k4_zf_refle,redefinition_k5_numbers,redefinition_k5_zf_refle,redefinition_k6_subset_1,redefinition_k6_zf_fund1,redefinition_k6_zf_lang1,redefinition_k8_funct_2,redefinition_k8_zf_fund1,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r2_wellord2,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r2_wellord2,s2_zf_refle__e13_10_2__zf_fund2,spc0_boole,spc10_boole,spc10_numerals,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,spc4_boole,spc4_numerals,spc5_boole,spc5_numerals,symmetry_r1_xboole_0,symmetry_r2_wellord2,t11_relset_1,t11_zfmodel2,t13_zf_model,t13_zfmodel2,t144_relat_1,t146_relat_1,t14_zfmodel2,t15_zf_model,t18_nat_1,t198_zf_lang1,t19_ordinal1,t19_zfmodel1,t1_arithm,t1_boole,t1_classes2,t1_numerals,t1_real,t1_subset,t1_xboole_1,t201_zf_lang1,t21_card_1,t22_ordinal1,t23_ordinal1,t24_zf_refle,t25_zf_refle,t27_ordinal2,t28_zf_refle,t29_nat_1,t29_zf_refle,t2_classes1,t2_real,t2_subset,t2_tarski,t30_zfrefle1,t3_boole,t3_card_2,t3_real,t3_subset,t3_xboole_0,t4_boole,t4_real,t4_subset,t4_zfmodel2,t5_real,t5_subset,t6_boole,t6_real,t6_zfmodel2,t79_zf_lang1,t7_boole,t7_real,t7_xboole_1,t7_zfmodel2,t82_zf_lang1,t83_card_1,t84_card_1,t86_zf_lang1,t8_boole,t8_real,t8_xreal_1,t8_zf_refle,t9_xreal_1,t9_zfmodel2]), [file(zf_fund2,t3_zf_fund2)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). 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_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_finsub_1,axiom,( ! [A] : ( v4_finsub_1(A) => ( v1_finsub_1(A) & v3_finsub_1(A) ) ) ), file(finsub_1,cc1_finsub_1), []). 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_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(cc2_finsub_1,axiom,( ! [A] : ( ( v1_finsub_1(A) & v3_finsub_1(A) ) => v4_finsub_1(A) ) ), file(finsub_1,cc2_finsub_1), []). 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_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). 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_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_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(commutativity_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), []). 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(d1_enumset1,axiom,( ! [A,B,C,D] : ( D = k1_enumset1(A,B,C) <=> ! [E] : ( r2_hidden(E,D) <=> ~ ( E != A & E != B & E != C ) ) ) ), file(enumset1,d1_enumset1), []). 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_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_zf_fund2,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) ) => ( ( r2_hidden(k2_zf_lang(0),k6_zf_fund1(A)) => k1_zf_fund2(A,B,C) = a_3_0_zf_fund2(A,B,C) ) & ( ~ r2_hidden(k2_zf_lang(0),k6_zf_fund1(A)) => k1_zf_fund2(A,B,C) = k1_xboole_0 ) ) ) ) ) ), file(zf_fund2,d1_zf_fund2), []). 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(d2_ordinal1,axiom,( ! [A] : ( v1_ordinal1(A) <=> ! [B] : ( r2_hidden(B,A) => r1_tarski(B,A) ) ) ), file(ordinal1,d2_ordinal1), []). 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_xboole_0,axiom,( ! [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), []). fof(d2_zf_fund2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ( v1_zf_fund2(A) <=> ! [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) ) => ( r2_hidden(C,A) => r2_hidden(k1_zf_fund2(B,C,D),A) ) ) ) ) ) ) ), file(zf_fund2,d2_zf_fund2), []). 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(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d4_xboole_0,axiom,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), []). 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_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(dt_k11_zf_model,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_zf_lang(k11_zf_model(A)) & m2_finseq_1(k11_zf_model(A),k5_numbers) ) ) ), file(zf_model,k11_zf_model), []). fof(dt_k12_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_zf_lang(k12_zf_lang(A,B)) & m2_finseq_1(k12_zf_lang(A,B),k5_numbers) ) ) ), file(zf_lang,k12_zf_lang), []). fof(dt_k13_zf_lang,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & v1_zf_lang(B) & m1_finseq_1(B,k5_numbers) ) => ( v1_zf_lang(k13_zf_lang(A,B)) & m2_finseq_1(k13_zf_lang(A,B),k5_numbers) ) ) ), file(zf_lang,k13_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_enumset1,axiom,( $true ), file(enumset1,k1_enumset1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zf_fund2,axiom,( ! [A,B,C] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,B) & m1_relset_1(C,k1_zf_lang,B) ) => m1_subset_1(k1_zf_fund2(A,B,C),k1_zfmisc_1(B)) ) ), file(zf_fund2,k1_zf_fund2), []). 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_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k1_zfmodel1,axiom,( ! [A,B,C] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,B) & m1_relset_1(C,k1_zf_lang,B) ) => ( v1_funct_1(k1_zfmodel1(A,B,C)) & v1_funct_2(k1_zfmodel1(A,B,C),B,B) & m2_relset_1(k1_zfmodel1(A,B,C),B,B) ) ) ), file(zfmodel1,k1_zfmodel1), []). fof(dt_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ), file(funct_2,k2_funct_2), []). 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_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_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zf_lang,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_subset_1(k2_zf_lang(A),k5_numbers,k1_zf_lang) ) ), file(zf_lang,k2_zf_lang), []). 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_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_card_1,axiom,( $true ), file(card_1,k3_card_1), []). fof(dt_k3_card_3,axiom,( $true ), file(card_3,k3_card_3), []). fof(dt_k3_zf_lang1,axiom,( $true ), file(zf_lang1,k3_zf_lang1), []). 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_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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), []). fof(dt_k4_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), []). 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_lang1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( ~ v1_xboole_0(k4_zf_lang1(A)) & m1_subset_1(k4_zf_lang1(A),k1_zfmisc_1(k1_zf_lang)) ) ) ), file(zf_lang1,k4_zf_lang1), []). 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_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_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_lang1,axiom,( ! [A,B,C] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & m1_subset_1(B,k1_zf_lang) & m1_subset_1(C,k1_zf_lang) ) => ( v1_relat_1(k5_zf_lang1(A,B,C)) & v1_funct_1(k5_zf_lang1(A,B,C)) ) ) ), file(zf_lang1,k5_zf_lang1), []). 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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k6_subset_1), []). fof(dt_k6_zf_fund1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => ( v1_finset_1(k6_zf_fund1(A)) & m1_subset_1(k6_zf_fund1(A),k1_zfmisc_1(k1_zf_lang)) ) ) ), file(zf_fund1,k6_zf_fund1), []). fof(dt_k6_zf_lang1,axiom,( ! [A,B,C] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & m1_subset_1(B,k1_zf_lang) & m1_subset_1(C,k1_zf_lang) ) => ( v1_zf_lang(k6_zf_lang1(A,B,C)) & m2_finseq_1(k6_zf_lang1(A,B,C),k5_numbers) ) ) ), file(zf_lang1,k6_zf_lang1), []). 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_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_zf_fund1,axiom,( ! [A] : ( m1_subset_1(A,k1_zf_lang) => ( v1_finset_1(k8_zf_fund1(A)) & m1_subset_1(k8_zf_fund1(A),k1_zfmisc_1(k1_zf_lang)) ) ) ), file(zf_fund1,k8_zf_fund1), []). 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_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_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_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_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(fc12_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), []). fof(fc13_finset_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(B) ) => v1_finset_1(k9_relat_1(A,B)) ) ), file(finset_1,fc13_finset_1), []). fof(fc14_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). fof(fc1_card_1,axiom,( ! [A] : ( v3_ordinal1(A) => ( v1_ordinal1(k3_card_1(A)) & v2_ordinal1(k3_card_1(A)) & v3_ordinal1(k3_card_1(A)) & v1_card_1(k3_card_1(A)) ) ) ), file(card_1,fc1_card_1), []). fof(fc1_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_finsub_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_zfmisc_1(A)) & v1_finsub_1(k1_zfmisc_1(A)) & v3_finsub_1(k1_zfmisc_1(A)) & v4_finsub_1(k1_zfmisc_1(A)) ) ), file(finsub_1,fc1_finsub_1), []). 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_zf_lang,axiom,( ~ v1_xboole_0(k1_zf_lang) ), file(zf_lang,fc1_zf_lang), []). 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_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_finset_1,axiom,( ! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) ), file(finset_1,fc3_finset_1), []). 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(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(fc9_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), []). fof(fraenkel_a_3_0_zf_fund2,axiom,( ! [A,B,C,D] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) & ~ v1_xboole_0(C) & v1_funct_1(D) & v1_funct_2(D,k1_zf_lang,C) & m2_relset_1(D,k1_zf_lang,C) ) => ( r2_hidden(A,a_3_0_zf_fund2(B,C,D)) <=> ? [E] : ( m1_subset_1(E,C) & A = E & r1_zf_model(C,k1_zf_lang1(C,D,k2_zf_lang(0),E),B) ) ) ) ), file(zf_fund2,a_3_0_zf_fund2), []). fof(fraenkel_a_7_0_zf_fund2,axiom,( ! [A,B,C,D,E,F,G,H] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) & v2_relat_1(C) & v1_zf_refle(C,B) & m1_ordinal1(C,B) & v1_zf_lang(D) & m2_finseq_1(D,k5_numbers) & m2_subset_1(E,k1_numbers,k5_numbers) & m1_ordinal4(F,B) & v1_funct_1(G) & v1_funct_2(G,k1_zf_lang,k5_zf_refle(B,C,F)) & m2_relset_1(G,k1_zf_lang,k5_zf_refle(B,C,F)) & m1_subset_1(H,k5_zf_refle(B,C,F)) ) => ( r2_hidden(A,a_7_0_zf_fund2(B,C,D,E,F,G,H)) <=> ? [I] : ( m1_subset_1(I,k5_zf_refle(B,C,F)) & A = I & r1_zf_model(k5_zf_refle(B,C,F),k1_zf_lang1(k5_zf_refle(B,C,F),k1_zf_lang1(k5_zf_refle(B,C,F),k1_zf_lang1(k5_zf_refle(B,C,F),G,k2_zf_lang(0),k8_funct_2(k1_zf_lang,k5_zf_refle(B,C,F),G,k2_zf_lang(4))),k2_zf_lang(k1_nat_1(E,10)),H),k2_zf_lang(0),I),k13_zf_lang(k2_zf_lang(3),k7_zf_lang(k5_zf_lang(k2_zf_lang(3),k2_zf_lang(k1_nat_1(E,10))),k6_zf_lang1(k6_zf_lang1(D,k2_zf_lang(0),k2_zf_lang(k1_nat_1(E,5))),k2_zf_lang(4),k2_zf_lang(0))))) ) ) ) ), file(zf_fund2,a_7_0_zf_fund2), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_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(idempotence_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), []). fof(l4_zf_fund2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ~ ( r2_hidden(B,k3_card_3(A)) & ! [C] : ~ ( r2_hidden(C,k1_relat_1(A)) & r2_hidden(B,k1_funct_1(A,C)) ) ) ) ), file(zf_fund2,l4_zf_fund2), []). fof(l6_zf_fund2,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] : ( m2_subset_1(D,k5_numbers,k1_zf_lang) => ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(0),k2_zf_lang(1),k2_zf_lang(2)),k6_zf_fund1(A)) & r1_zf_model(B,C,k8_zf_lang(k2_zf_lang(3),k13_zf_lang(k2_zf_lang(0),k8_zf_lang(k2_zf_lang(4),k12_zf_lang(A,k4_zf_lang(k2_zf_lang(4),k2_zf_lang(0))))))) ) => ( r2_hidden(D,k4_zf_lang1(A)) | ( r1_xboole_0(k1_enumset1(k2_zf_lang(0),k2_zf_lang(1),k2_zf_lang(2)),k6_zf_fund1(k6_zf_lang1(A,k2_zf_lang(0),D))) & r1_zf_model(B,C,k8_zf_lang(k2_zf_lang(3),k13_zf_lang(k2_zf_lang(0),k8_zf_lang(k2_zf_lang(4),k12_zf_lang(k6_zf_lang1(A,k2_zf_lang(0),D),k4_zf_lang(k2_zf_lang(4),k2_zf_lang(0))))))) & k1_zfmodel1(A,B,C) = k1_zfmodel1(k6_zf_lang1(A,k2_zf_lang(0),D),B,C) ) ) ) ) ) ) ) ), file(zf_fund2,l6_zf_fund2), []). fof(l7_zf_fund2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,B) => ! [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) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k1_zf_lang,B) & m2_relset_1(F,k1_zf_lang,B) ) => ! [G] : ( m2_subset_1(G,k5_numbers,k1_zf_lang) => ( ( E = F & D = C ) => k1_zf_lang1(A,E,G,D) = k1_zf_lang1(B,F,G,C) ) ) ) ) ) ) ) ) ), file(zf_fund2,l7_zf_fund2), []). fof(l8_zf_fund2,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) ) => ( r1_zf_model(B,C,k8_zf_lang(k2_zf_lang(3),k13_zf_lang(k2_zf_lang(0),k8_zf_lang(k2_zf_lang(4),k12_zf_lang(A,k4_zf_lang(k2_zf_lang(4),k2_zf_lang(0))))))) => ( r2_hidden(k2_zf_lang(4),k6_zf_fund1(A)) | ! [D] : ( m1_subset_1(D,B) => k2_funct_2(B,B,k1_zfmodel1(A,B,C),D) = k1_xboole_0 ) ) ) ) ) ) ), file(zf_fund2,l8_zf_fund2), []). fof(l9_zf_fund2,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ~ ( ~ r2_hidden(B,k4_zf_lang1(A)) & ~ r2_hidden(C,k4_zf_lang1(A)) & B != k2_zf_lang(0) & C != k2_zf_lang(0) & C != k2_zf_lang(4) & ~ ( r2_hidden(k2_zf_lang(4),k6_zf_fund1(A)) <=> r2_hidden(k2_zf_lang(0),k6_zf_fund1(k13_zf_lang(k2_zf_lang(3),k7_zf_lang(k5_zf_lang(k2_zf_lang(3),B),k6_zf_lang1(k6_zf_lang1(A,k2_zf_lang(0),C),k2_zf_lang(4),k2_zf_lang(0)))))) ) ) ) ) ) ), file(zf_fund2,l9_zf_fund2), []). 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_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). 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_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(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_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), []). 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(rc3_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). 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_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ), file(funct_2,k2_funct_2), []). 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_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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), []). fof(redefinition_k4_zf_lang1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => k4_zf_lang1(A) = k3_zf_lang1(A) ) ), file(zf_lang1,k4_zf_lang1), []). 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_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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) ), file(subset_1,k6_subset_1), []). fof(redefinition_k6_zf_fund1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) ) => k6_zf_fund1(A) = k1_zf_model(A) ) ), file(zf_fund1,k6_zf_fund1), []). fof(redefinition_k6_zf_lang1,axiom,( ! [A,B,C] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & m1_subset_1(B,k1_zf_lang) & m1_subset_1(C,k1_zf_lang) ) => k6_zf_lang1(A,B,C) = k5_zf_lang1(A,B,C) ) ), file(zf_lang1,k6_zf_lang1), []). 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_k8_zf_fund1,axiom,( ! [A] : ( m1_subset_1(A,k1_zf_lang) => k8_zf_fund1(A) = k1_tarski(A) ) ), file(zf_fund1,k8_zf_fund1), []). 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_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_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) <=> r2_tarski(A,B) ) ), file(wellord2,r2_wellord2), []). 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(s2_zf_refle__e13_10_2__zf_fund2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & v2_relat_1(B) & v1_zf_refle(B,A) & m1_ordinal1(B,A) ) => ( ! [C] : ( m1_subset_1(C,k4_zf_refle(A,B)) => ? [D] : ( m1_ordinal4(D,A) & r2_hidden(C,k1_funct_1(B,D)) ) ) => ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & k1_relat_1(C) = k4_zf_refle(A,B) & ! [D] : ( m1_subset_1(D,k4_zf_refle(A,B)) => ? [E] : ( m1_ordinal4(E,A) & E = k1_funct_1(C,D) & r2_hidden(D,k1_funct_1(B,E)) & ! [F] : ( m1_ordinal4(F,A) => ( r2_hidden(D,k1_funct_1(B,F)) => r1_tarski(E,F) ) ) ) ) ) ) ) ), file(zf_fund2,s2_zf_refle__e13_10_2__zf_fund2), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc10_boole,axiom,( ~ v1_xboole_0(10) ), file(boole,spc10_boole), []). fof(spc10_numerals,axiom, ( v2_xreal_0(10) & m1_subset_1(10,k5_numbers) ), file(numerals,spc10_numerals), []). 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_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(symmetry_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) => r2_wellord2(B,A) ) ), file(wellord2,r2_wellord2), []). 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(t11_zfmodel2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,A) & m2_relset_1(C,k1_zf_lang,A) ) => ( r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(3),k13_zf_lang(k2_zf_lang(0),k8_zf_lang(k2_zf_lang(4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(4),k2_zf_lang(0))))))) => ( r2_hidden(k2_zf_lang(0),k2_zf_model(B)) | ! [D] : ( m1_subset_1(D,A) => ! [E] : ( m1_subset_1(E,A) => ( k8_funct_2(A,A,k1_zfmodel1(B,A,C),D) = E <=> r1_zf_model(A,k1_zf_lang1(A,k1_zf_lang1(A,C,k2_zf_lang(3),D),k2_zf_lang(4),E),B) ) ) ) ) ) ) ) ) ), file(zfmodel2,t11_zfmodel2), []). 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(t13_zfmodel2,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( ( v1_zf_lang(D) & m2_finseq_1(D,k5_numbers) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,C) & m2_relset_1(E,k1_zf_lang,C) ) => ( ~ r2_hidden(A,k4_zf_lang1(D)) => ( r1_zf_model(C,E,k6_zf_lang1(D,B,A)) <=> r1_zf_model(C,k1_zf_lang1(C,E,B,k8_funct_2(k1_zf_lang,C,E,A)),D) ) ) ) ) ) ) ) ), file(zfmodel2,t13_zfmodel2), []). fof(t144_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k9_relat_1(B,A),k2_relat_1(B)) ) ), file(relat_1,t144_relat_1), []). fof(t146_relat_1,axiom,( ! [A] : ( v1_relat_1(A) => k9_relat_1(A,k1_relat_1(A)) = k2_relat_1(A) ) ), file(relat_1,t146_relat_1), []). fof(t14_zfmodel2,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( ( v1_zf_lang(D) & m2_finseq_1(D,k5_numbers) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,C) & m2_relset_1(E,k1_zf_lang,C) ) => ( r1_zf_model(C,E,D) => ( r2_hidden(A,k4_zf_lang1(D)) | r1_zf_model(C,k1_zf_lang1(C,E,A,k8_funct_2(k1_zf_lang,C,E,B)),k6_zf_lang1(D,B,A)) ) ) ) ) ) ) ) ), file(zfmodel2,t14_zfmodel2), []). 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(t18_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_1), []). fof(t198_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] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ~ ( B != C & r2_hidden(B,k4_zf_lang1(k6_zf_lang1(A,B,C))) ) ) ) ) ), file(zf_lang1,t198_zf_lang1), []). 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(t19_zfmodel1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ( v1_ordinal1(A) => ( ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => ( r1_xboole_0(k1_enumset1(k2_zf_lang(0),k2_zf_lang(1),k2_zf_lang(2)),k2_zf_model(B)) => r2_zf_model(A,k11_zf_model(B)) ) ) <=> ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,A) & m2_relset_1(C,k1_zf_lang,A) ) => ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(0),k2_zf_lang(1),k2_zf_lang(2)),k2_zf_model(B)) & r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(3),k13_zf_lang(k2_zf_lang(0),k8_zf_lang(k2_zf_lang(4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(4),k2_zf_lang(0))))))) ) => ! [D] : ( m1_subset_1(D,A) => r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) ) ) ) ), file(zfmodel1,t19_zfmodel1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). 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_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). 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(t201_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] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => r1_tarski(k4_zf_lang1(k6_zf_lang1(A,B,C)),k2_xboole_0(k4_xboole_0(k4_zf_lang1(A),k1_tarski(B)),k1_tarski(C))) ) ) ) ), file(zf_lang1,t201_zf_lang1), []). 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(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_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(t27_ordinal2,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( r2_hidden(A,B) => r2_hidden(A,k7_ordinal2(B)) ) ) ), file(ordinal2,t27_ordinal2), []). 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_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t29_zf_refle,axiom,( ! [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) ) ) ) ) ) ) ) ) ) ) ), file(zf_refle,t29_zf_refle), []). 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_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), []). 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(t30_zfrefle1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_ordinal4(B,A) => ! [C] : ( m2_ordinal4(C,A) => ~ ( r2_hidden(k5_ordinal2,A) & v2_ordinal2(C) & v3_ordinal2(C) & ! [D] : ( m1_ordinal4(D,A) => ~ ( r2_hidden(B,D) & k4_ordinal4(A,C,D) = D ) ) ) ) ) ) ), file(zfrefle1,t30_zfrefle1), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). 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_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_xboole_0,axiom,( ! [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), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). 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(t4_zfmodel2,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(k2_zf_lang(C),k4_zf_lang1(A)) & r1_xreal_0(B,C) ) ) ) & ~ ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => r2_hidden(B,k4_zf_lang1(A)) ) ) ) ), file(zfmodel2,t4_zfmodel2), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). 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(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t6_zfmodel2,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,B) => ! [D] : ( ( v1_zf_lang(D) & m2_finseq_1(D,k5_numbers) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,B) & m2_relset_1(E,k1_zf_lang,B) ) => ( ~ r2_hidden(A,k4_zf_lang1(D)) => ( r1_zf_model(B,E,D) <=> r1_zf_model(B,k1_zf_lang1(B,E,A,C),D) ) ) ) ) ) ) ) ), file(zfmodel2,t6_zfmodel2), []). fof(t79_zf_lang1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m1_subset_1(D,C) => ! [E] : ( m1_subset_1(E,C) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k1_zf_lang,C) & m2_relset_1(F,k1_zf_lang,C) ) => ( A != B => k1_zf_lang1(C,k1_zf_lang1(C,F,A,D),B,E) = k1_zf_lang1(C,k1_zf_lang1(C,F,B,E),A,D) ) ) ) ) ) ) ) ), file(zf_lang1,t79_zf_lang1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t7_xboole_1,axiom,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), []). fof(t7_zfmodel2,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( m1_subset_1(E,D) => ! [F] : ( m1_subset_1(F,D) => ! [G] : ( m1_subset_1(G,D) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,k1_zf_lang,D) & m2_relset_1(H,k1_zf_lang,D) ) => ~ ( A != B & B != C & C != A & ~ ( k1_zf_lang1(D,k1_zf_lang1(D,k1_zf_lang1(D,H,A,E),B,F),C,G) = k1_zf_lang1(D,k1_zf_lang1(D,k1_zf_lang1(D,H,C,G),B,F),A,E) & k1_zf_lang1(D,k1_zf_lang1(D,k1_zf_lang1(D,H,A,E),B,F),C,G) = k1_zf_lang1(D,k1_zf_lang1(D,k1_zf_lang1(D,H,B,F),C,G),A,E) ) ) ) ) ) ) ) ) ) ) ), file(zfmodel2,t7_zfmodel2), []). fof(t82_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,k13_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,t82_zf_lang1), []). fof(t83_card_1,axiom,( k3_card_1(0) = k5_ordinal2 ), file(card_1,t83_card_1), []). fof(t84_card_1,axiom,( k1_card_1(k5_ordinal2) = k5_ordinal2 ), file(card_1,t84_card_1), []). fof(t86_zf_lang1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( k2_zf_lang(A) = k2_zf_lang(B) => A = B ) ) ) ), file(zf_lang1,t86_zf_lang1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t8_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). 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(t9_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), []). fof(t9_zfmodel2,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => ! [D] : ( m2_subset_1(D,k5_numbers,k1_zf_lang) => ! [E] : ( ~ v1_xboole_0(E) => ! [F] : ( m1_subset_1(F,E) => ! [G] : ( m1_subset_1(G,E) => ! [H] : ( m1_subset_1(H,E) => ! [I] : ( m1_subset_1(I,E) => ! [J] : ( m1_subset_1(J,E) => ! [K] : ( ( v1_funct_1(K) & v1_funct_2(K,k1_zf_lang,E) & m2_relset_1(K,k1_zf_lang,E) ) => ( k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,K,A,F),B,G),A,H) = k1_zf_lang1(E,k1_zf_lang1(E,K,B,G),A,H) & k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,K,A,F),B,G),C,I),A,H) = k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,K,B,G),C,I),A,H) & k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,K,A,F),B,G),C,I),D,J),A,H) = k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,k1_zf_lang1(E,K,B,G),C,I),D,J),A,H) ) ) ) ) ) ) ) ) ) ) ) ) ), file(zfmodel2,t9_zfmodel2), []).