% Mizar problem: t18_zf_fund1,zf_fund1,1488,60 fof(t18_zf_fund1,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( ~ v1_xboole_0(C) => ( ( v8_zf_fund1(B,A) & r2_hidden(C,B) ) => ! [D] : ( m2_subset_1(D,k5_numbers,k1_zf_lang) => ! [E] : ( m2_subset_1(E,k5_numbers,k1_zf_lang) => ( r2_hidden(k10_zf_fund1(k4_zf_lang(D,E),C),B) & r2_hidden(k10_zf_fund1(k5_zf_lang(D,E),C),B) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_card_1,cc1_classes2,cc1_finset_1,cc1_funct_1,cc1_ordinal1,cc1_relset_1,cc2_card_1,cc2_classes2,cc2_finset_1,cc2_funct_1,cc2_ordinal1,cc3_card_1,cc3_classes2,cc3_ordinal1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k4_subset_1,commutativity_k9_zf_fund1,connectedness_r1_ordinal1,d10_xboole_0,d13_zf_fund1,d1_tarski,d2_funct_2,d2_tarski,d3_tarski,d3_zf_fund1,d4_ordinal2,d4_zf_fund1,d5_tarski,d5_zf_fund1,d6_zf_fund1,d9_zf_fund1,dt_k10_zf_fund1,dt_k18_zf_lang,dt_k19_zf_lang,dt_k1_classes2,dt_k1_funct_1,dt_k1_funct_2,dt_k1_numbers,dt_k1_ordinal1,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zf_fund1,dt_k1_zf_lang,dt_k1_zf_model,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_funct_1,dt_k2_ordinal4,dt_k2_relat_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zf_fund1,dt_k2_zf_lang,dt_k2_zf_model,dt_k2_zfmisc_1,dt_k3_ordinal4,dt_k3_tarski,dt_k3_zf_fund1,dt_k3_zf_model,dt_k4_ordinal2,dt_k4_subset_1,dt_k4_tarski,dt_k4_zf_fund1,dt_k4_zf_lang,dt_k4_zf_model,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relat_1,dt_k5_zf_fund1,dt_k5_zf_lang,dt_k5_zf_model,dt_k6_classes2,dt_k6_zf_fund1,dt_k6_zf_lang,dt_k7_funct_2,dt_k7_relat_1,dt_k7_zf_fund1,dt_k7_zf_lang,dt_k8_funct_2,dt_k8_zf_fund1,dt_k8_zf_lang,dt_k9_relat_1,dt_k9_zf_fund1,dt_m1_finseq_1,dt_m1_ordinal4,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,existence_m1_finseq_1,existence_m1_ordinal4,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,fc13_finset_1,fc14_finset_1,fc1_finset_1,fc1_funct_1,fc1_ordinal1,fc1_ordinal2,fc1_zf_lang,fc2_finset_1,fc2_ordinal1,fc2_zf_lang,fc3_ordinal1,fc3_zf_lang,fc4_funct_1,fc4_ordinal1,fc4_ordinal2,fc4_zf_lang,fc5_zf_lang,fc6_zf_lang,fc9_finset_1,fraenkel_a_2_0_zf_fund1,fraenkel_a_2_2_zf_fund1,fraenkel_a_3_0_zf_fund1,fraenkel_a_3_8_zf_fund1,fraenkel_a_4_4_zf_fund1,idempotence_k2_xboole_0,idempotence_k4_subset_1,l16_zf_fund1,l19_zf_fund1,l20_zf_fund1,l23_zf_fund1,l37_zf_fund1,l46_zf_fund1,l47_zf_fund1,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,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc4_finset_1,rc4_funct_1,redefinition_k1_classes2,redefinition_k2_zf_fund1,redefinition_k2_zf_model,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k5_zf_model,redefinition_k6_classes2,redefinition_k6_zf_fund1,redefinition_k7_funct_2,redefinition_k7_zf_fund1,redefinition_k8_funct_2,redefinition_k8_zf_fund1,redefinition_k9_zf_fund1,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_ordinal1,reflexivity_r1_ordinal1,reflexivity_r1_tarski,s10_funct_2__e9_45_1_2_5_1__zf_fund1,t10_zf_fund1,t13_funcop_1,t15_grfunc_1,t16_zf_lang,t17_zf_fund1,t18_grfunc_1,t19_grfunc_1,t19_ordinal2,t1_boole,t1_subset,t1_zf_fund1,t1_zf_lang1,t23_relat_1,t2_subset,t2_tarski,t2_zf_fund1,t2_zf_lang1,t33_grfunc_1,t35_ordinal4,t38_zfmisc_1,t3_subset,t3_zf_fund1,t41_enumset1,t41_ordinal1,t4_subset,t57_funcop_1,t5_subset,t63_zf_lang1,t64_zf_lang1,t69_enumset1,t6_boole,t6_zf_fund1,t7_boole,t7_zf_fund1,t7_zf_model,t8_boole,t8_zf_model]), [file(zf_fund1,t18_zf_fund1)]). 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_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_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_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_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). 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_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(commutativity_k9_zf_fund1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => k9_zf_fund1(A,B) = k9_zf_fund1(B,A) ) ), file(zf_fund1,k9_zf_fund1), []). 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(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d13_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v8_zf_fund1(B,A) <=> ( v1_zf_fund1(B,A) & v2_zf_fund1(B,A) & v3_zf_fund1(B,A) & v4_zf_fund1(B,A) & v5_zf_fund1(B,A) & v6_zf_fund1(B,A) & v7_zf_fund1(B,A) ) ) ) ) ), file(zf_fund1,d13_zf_fund1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). 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_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). 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_zf_fund1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k4_zf_fund1(A) <=> k2_zf_lang(B) = A ) ) ) ), file(zf_fund1,d3_zf_fund1), []). fof(d4_ordinal2,axiom,( k4_ordinal2 = k1_ordinal1(k1_xboole_0) ), file(ordinal2,d4_ordinal2), []). fof(d4_zf_fund1,axiom,( ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_zf_lang)) ) => k5_zf_fund1(A) = k9_relat_1(k2_funct_1(k3_zf_fund1),A) ) ), file(zf_fund1,d4_zf_fund1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d5_zf_fund1,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( C = k10_zf_fund1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,B) & m2_relset_1(E,k1_zf_lang,B) & D = k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,E),k5_zf_fund1(k6_zf_fund1(A))) & r2_hidden(E,k5_zf_model(A,B)) ) ) ) ) ) ), file(zf_fund1,d5_zf_fund1), []). fof(d6_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v1_zf_fund1(B,A) <=> ! [C] : ( m1_subset_1(C,A) => ( r2_hidden(C,B) => r2_hidden(a_2_0_zf_fund1(A,C),B) ) ) ) ) ) ), file(zf_fund1,d6_zf_fund1), []). fof(d9_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v4_zf_fund1(B,A) <=> ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => ( ( r2_hidden(C,B) & r2_hidden(D,B) ) => r2_hidden(a_3_0_zf_fund1(A,C,D),B) ) ) ) ) ) ) ), file(zf_fund1,d9_zf_fund1), []). fof(dt_k10_zf_fund1,axiom,( $true ), file(zf_fund1,k10_zf_fund1), []). 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_classes2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) ) => m1_subset_1(k1_classes2(A,B),A) ) ), file(classes2,k1_classes2), []). 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_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zf_fund1,axiom,( $true ), file(zf_fund1,k1_zf_fund1), []). 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_model,axiom,( $true ), file(zf_model,k1_zf_model), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k2_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), []). fof(dt_k2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k2_funct_1(A)) & v1_funct_1(k2_funct_1(A)) ) ) ), file(funct_1,k2_funct_1), []). 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_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zf_fund1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k2_zf_fund1(A,B,C),A) ) ), file(zf_fund1,k2_zf_fund1), []). 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_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_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_zf_fund1,axiom, ( v1_funct_1(k3_zf_fund1) & v1_funct_2(k3_zf_fund1,k5_ordinal2,k1_zf_lang) & m2_relset_1(k3_zf_fund1,k5_ordinal2,k1_zf_lang) ), file(zf_fund1,k3_zf_fund1), []). fof(dt_k3_zf_model,axiom,( $true ), file(zf_model,k3_zf_model), []). fof(dt_k4_ordinal2,axiom, ( v3_ordinal1(k4_ordinal2) & ~ v1_xboole_0(k4_ordinal2) ), file(ordinal2,k4_ordinal2), []). 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_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_zf_fund1,axiom,( ! [A] : ( m1_subset_1(A,k1_zf_lang) => m2_subset_1(k4_zf_fund1(A),k1_numbers,k5_numbers) ) ), file(zf_fund1,k4_zf_fund1), []). 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_model,axiom,( $true ), file(zf_model,k4_zf_model), []). 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_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_fund1,axiom,( ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_zf_lang)) ) => ( v1_finset_1(k5_zf_fund1(A)) & m1_subset_1(k5_zf_fund1(A),k1_zfmisc_1(k5_ordinal2)) ) ) ), file(zf_fund1,k5_zf_fund1), []). 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_model,axiom,( ! [A,B] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & ~ v1_xboole_0(B) ) => m1_subset_1(k5_zf_model(A,B),k1_zfmisc_1(k3_zf_model(B))) ) ), file(zf_model,k5_zf_model), []). fof(dt_k6_classes2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k6_classes2(A,B,C),A) ) ), file(classes2,k6_classes2), []). 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_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_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => ( v1_funct_1(k7_funct_2(A,B,C,D,E)) & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C) & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ), file(funct_2,k7_funct_2), []). 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_fund1,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_finset_1(k7_zf_fund1(A)) & m1_subset_1(k7_zf_fund1(A),k1_zfmisc_1(k5_ordinal2)) ) ) ), file(zf_fund1,k7_zf_fund1), []). 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_k9_zf_fund1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => ( v1_finset_1(k9_zf_fund1(A,B)) & m1_subset_1(k9_zf_fund1(A,B),k1_zfmisc_1(k1_zf_lang)) ) ) ), file(zf_fund1,k9_zf_fund1), []). 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_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_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_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_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(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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_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_zf_lang,axiom,( ~ v1_xboole_0(k1_zf_lang) ), file(zf_lang,fc1_zf_lang), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_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_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_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_ordinal2,axiom,( ! [A,B] : ( v3_ordinal1(B) => ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) & v1_ordinal2(k2_funcop_1(A,B)) ) ) ), file(ordinal2,fc4_ordinal2), []). 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_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_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_2_0_zf_fund1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) & m1_subset_1(C,B) ) => ( r2_hidden(A,a_2_0_zf_fund1(B,C)) <=> ? [D,E] : ( m1_subset_1(D,B) & m1_subset_1(E,B) & A = k2_tarski(k4_tarski(k2_ordinal4(B),D),k4_tarski(k3_ordinal4(B),E)) & r2_hidden(D,E) & r2_hidden(D,C) & r2_hidden(E,C) ) ) ) ), file(zf_fund1,a_2_0_zf_fund1), []). fof(fraenkel_a_2_2_zf_fund1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) & m1_subset_1(C,B) ) => ( r2_hidden(A,a_2_2_zf_fund1(B,C)) <=> ? [D] : ( m1_subset_1(D,B) & A = k2_tarski(k4_tarski(k2_ordinal4(B),D),k4_tarski(k3_ordinal4(B),D)) & r2_hidden(D,C) ) ) ) ), file(zf_fund1,a_2_2_zf_fund1), []). fof(fraenkel_a_3_0_zf_fund1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) & m1_subset_1(C,B) & m1_subset_1(D,B) ) => ( r2_hidden(A,a_3_0_zf_fund1(B,C,D)) <=> ? [E,F] : ( m1_subset_1(E,B) & m1_subset_1(F,B) & A = k1_classes2(B,k6_classes2(B,E,F)) & r2_hidden(E,C) & r2_hidden(F,D) ) ) ) ), file(zf_fund1,a_3_0_zf_fund1), []). fof(fraenkel_a_3_8_zf_fund1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) & m1_subset_1(C,B) & m1_subset_1(D,B) ) => ( r2_hidden(A,a_3_8_zf_fund1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,B) & A = k2_zf_fund1(B,C,E) & r2_hidden(E,D) ) ) ) ), file(zf_fund1,a_3_8_zf_fund1), []). fof(fraenkel_a_4_4_zf_fund1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_classes2(B) & m2_subset_1(C,k5_numbers,k1_zf_lang) & m2_subset_1(D,k5_numbers,k1_zf_lang) & m1_subset_1(E,B) ) => ( r2_hidden(A,a_4_4_zf_fund1(B,C,D,E)) <=> ? [F,G] : ( m1_subset_1(F,B) & m1_subset_1(G,B) & A = k1_classes2(B,k6_classes2(B,F,G)) & r2_hidden(F,k5_zf_fund1(k6_zf_fund1(k4_zf_lang(C,D)))) & r2_hidden(G,E) ) ) ) ), file(zf_fund1,a_4_4_zf_fund1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). 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(l16_zf_fund1,axiom,( ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(k5_ordinal2)) ) => ! [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) ) => ( k1_relat_1(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C),A)) = A & r1_tarski(k2_relat_1(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C),A)),B) & r2_hidden(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C),A),k1_funct_2(A,B)) & k1_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C)) = k5_ordinal2 & r1_tarski(k2_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C)),B) ) ) ) ) ), file(zf_fund1,l16_zf_fund1), []). fof(l19_zf_fund1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => k5_zf_fund1(k8_zf_fund1(A)) = k7_zf_fund1(k4_zf_fund1(A)) ) ), file(zf_fund1,l19_zf_fund1), []). fof(l20_zf_fund1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => k5_zf_fund1(k9_zf_fund1(A,B)) = k2_tarski(k4_zf_fund1(A),k4_zf_fund1(B)) ) ) ), file(zf_fund1,l20_zf_fund1), []). fof(l23_zf_fund1,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] : ( ( v1_zf_lang(D) & m2_finseq_1(D,k5_numbers) ) => ( r2_hidden(C,k6_zf_fund1(D)) => k1_funct_1(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,A,k3_zf_fund1,B),k5_zf_fund1(k6_zf_fund1(D))),k4_zf_fund1(C)) = k8_funct_2(k1_zf_lang,A,B,C) ) ) ) ) ) ), file(zf_fund1,l23_zf_fund1), []). fof(l37_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v8_zf_fund1(B,A) => ( r2_hidden(k2_ordinal4(A),B) & r2_hidden(k3_ordinal4(A),B) ) ) ) ) ), file(zf_fund1,l37_zf_fund1), []). fof(l46_zf_fund1,axiom,( ! [A,B,C,D,E,F] : ( A != B => k1_zf_fund1(k2_tarski(k4_tarski(C,A),k4_tarski(D,B)),k2_tarski(k4_tarski(A,E),k4_tarski(B,F))) = k2_tarski(k4_tarski(C,E),k4_tarski(D,F)) ) ), file(zf_fund1,l46_zf_fund1), []). fof(l47_zf_fund1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( k1_relat_1(C) = k2_tarski(A,B) <=> C = k2_tarski(k4_tarski(A,k1_funct_1(C,A)),k4_tarski(B,k1_funct_1(C,B))) ) ) ), file(zf_fund1,l47_zf_fund1), []). 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(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(redefinition_k1_classes2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) ) => k1_classes2(A,B) = k1_tarski(B) ) ), file(classes2,k1_classes2), []). fof(redefinition_k2_zf_fund1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k2_zf_fund1(A,B,C) = k1_zf_fund1(B,C) ) ), file(zf_fund1,k2_zf_fund1), []). 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_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_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_zf_model,axiom,( ! [A,B] : ( ( v1_zf_lang(A) & m1_finseq_1(A,k5_numbers) & ~ v1_xboole_0(B) ) => k5_zf_model(A,B) = k4_zf_model(A,B) ) ), file(zf_model,k5_zf_model), []). fof(redefinition_k6_classes2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k6_classes2(A,B,C) = k4_tarski(B,C) ) ), file(classes2,k6_classes2), []). 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_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ), file(funct_2,k7_funct_2), []). fof(redefinition_k7_zf_fund1,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => k7_zf_fund1(A) = k1_tarski(A) ) ), file(zf_fund1,k7_zf_fund1), []). 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_k9_zf_fund1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_zf_lang) & m1_subset_1(B,k1_zf_lang) ) => k9_zf_fund1(A,B) = k2_tarski(A,B) ) ), file(zf_fund1,k9_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_r1_ordinal1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => ( r1_ordinal1(A,B) <=> r1_tarski(A,B) ) ) ), file(ordinal1,r1_ordinal1), []). 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(s10_funct_2__e9_45_1_2_5_1__zf_fund1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m2_subset_1(D,k5_numbers,k1_zf_lang) ) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k1_zf_lang,A) & m2_relset_1(E,k1_zf_lang,A) & k8_funct_2(k1_zf_lang,A,E,D) = B & ! [F] : ( m1_subset_1(F,k1_zf_lang) => ( F != D => k8_funct_2(k1_zf_lang,A,E,F) = C ) ) ) ) ), file(zf_fund1,s10_funct_2__e9_45_1_2_5_1__zf_fund1), []). fof(t10_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( ( ~ v1_xboole_0(D) & m1_subset_1(D,k1_zfmisc_1(A)) ) => ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(k5_ordinal2)) ) => ( ( v8_zf_fund1(D,A) & r2_hidden(B,k1_funct_2(E,k5_ordinal2)) & r2_hidden(C,D) ) => r2_hidden(a_3_8_zf_fund1(A,B,C),D) ) ) ) ) ) ) ), file(zf_fund1,t10_zf_fund1), []). fof(t13_funcop_1,axiom,( ! [A,B,C] : ( r2_hidden(B,A) => k1_funct_1(k2_funcop_1(A,C),B) = C ) ), file(funcop_1,t13_funcop_1), []). fof(t15_grfunc_1,axiom,( ! [A,B] : ( v1_relat_1(k1_tarski(k4_tarski(A,B))) & v1_funct_1(k1_tarski(k4_tarski(A,B))) ) ), file(grfunc_1,t15_grfunc_1), []). fof(t16_zf_lang,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ( ~ ( v2_zf_lang(A) & ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => A != k4_zf_lang(B,C) ) ) ) & ( ? [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) & ? [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) & A = k4_zf_lang(B,C) ) ) => v2_zf_lang(A) ) & ~ ( v3_zf_lang(A) & ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) => A != k5_zf_lang(B,C) ) ) ) & ( ? [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) & ? [C] : ( m2_subset_1(C,k5_numbers,k1_zf_lang) & A = k5_zf_lang(B,C) ) ) => v3_zf_lang(A) ) & ~ ( v4_zf_lang(A) & ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => A != k6_zf_lang(B) ) ) & ( ? [B] : ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) & A = k6_zf_lang(B) ) => v4_zf_lang(A) ) & ~ ( v5_zf_lang(A) & ! [B] : ( ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) ) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => A != k7_zf_lang(B,C) ) ) ) & ( ? [B] : ( v1_zf_lang(B) & m2_finseq_1(B,k5_numbers) & ? [C] : ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) & A = k7_zf_lang(B,C) ) ) => v5_zf_lang(A) ) & ~ ( v6_zf_lang(A) & ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ! [C] : ( ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) ) => A != k8_zf_lang(B,C) ) ) ) & ( ? [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) & ? [C] : ( v1_zf_lang(C) & m2_finseq_1(C,k5_numbers) & A = k8_zf_lang(B,C) ) ) => v6_zf_lang(A) ) ) ) ), file(zf_lang,t16_zf_lang), []). fof(t17_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(A)) ) => ( ( v8_zf_fund1(C,A) & r2_hidden(B,C) ) => r2_hidden(a_2_2_zf_fund1(A,B),C) ) ) ) ) ), file(zf_fund1,t17_zf_fund1), []). fof(t18_grfunc_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( k1_relat_1(B) = k1_tarski(A) => B = k1_tarski(k4_tarski(A,k1_funct_1(B,A))) ) ) ), file(grfunc_1,t18_grfunc_1), []). fof(t19_grfunc_1,axiom,( ! [A,B,C,D] : ( ( v1_relat_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D))) & v1_funct_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D))) ) <=> ( A = C => B = D ) ) ), file(grfunc_1,t19_grfunc_1), []). fof(t19_ordinal2,axiom, ( r2_hidden(k1_xboole_0,k5_ordinal2) & v4_ordinal1(k5_ordinal2) & ! [A] : ( v3_ordinal1(A) => ( ( r2_hidden(k1_xboole_0,A) & v4_ordinal1(A) ) => r1_ordinal1(k5_ordinal2,A) ) ) ), file(ordinal2,t19_ordinal2), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C,D] : ( r1_tarski(B,A) & ( r2_hidden(C,B) => m1_subset_1(C,A) ) & ( ( r2_hidden(C,D) & r2_hidden(D,B) ) => m1_subset_1(C,A) ) ) ) ) ), file(zf_fund1,t1_zf_fund1), []). fof(t1_zf_lang1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ( k18_zf_lang(k4_zf_lang(A,B)) = A & k19_zf_lang(k4_zf_lang(A,B)) = B ) ) ) ), file(zf_lang1,t1_zf_lang1), []). fof(t23_relat_1,axiom,( ! [A,B,C] : ( v1_relat_1(C) => ( C = k1_tarski(k4_tarski(A,B)) => ( k1_relat_1(C) = k1_tarski(A) & k2_relat_1(C) = k1_tarski(B) ) ) ) ), file(relat_1,t23_relat_1), []). 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(t2_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C,D] : ( v8_zf_fund1(B,A) => ( ( r2_hidden(C,B) => r2_hidden(k1_tarski(C),B) ) & ( r2_hidden(k1_tarski(C),B) => r2_hidden(C,B) ) & ( r2_hidden(D,B) => r2_hidden(k3_tarski(D),B) ) ) ) ) ) ), file(zf_fund1,t2_zf_fund1), []). fof(t2_zf_lang1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => ( k18_zf_lang(k5_zf_lang(A,B)) = A & k19_zf_lang(k5_zf_lang(A,B)) = B ) ) ) ), file(zf_lang1,t2_zf_lang1), []). fof(t33_grfunc_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( A = k2_xboole_0(B,C) => ( k1_relat_1(A) = k2_xboole_0(k1_relat_1(B),k1_relat_1(C)) & k2_relat_1(A) = k2_xboole_0(k2_relat_1(B),k2_relat_1(C)) ) ) ) ) ) ), file(grfunc_1,t33_grfunc_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(t38_zfmisc_1,axiom,( ! [A,B,C] : ( r1_tarski(k2_tarski(A,B),C) <=> ( r2_hidden(A,C) & r2_hidden(B,C) ) ) ), file(zfmisc_1,t38_zfmisc_1), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v8_zf_fund1(B,A) => r2_hidden(k1_xboole_0,B) ) ) ) ), file(zf_fund1,t3_zf_fund1), []). fof(t41_enumset1,axiom,( ! [A,B] : k2_tarski(A,B) = k2_xboole_0(k1_tarski(A),k1_tarski(B)) ), file(enumset1,t41_enumset1), []). 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(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(t57_funcop_1,axiom,( ! [A,B,C] : ( r2_hidden(C,B) => ( v1_funct_1(k2_funcop_1(A,C)) & v1_funct_2(k2_funcop_1(A,C),A,B) & m2_relset_1(k2_funcop_1(A,C),A,B) ) ) ), file(funcop_1,t57_funcop_1), []). 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(t63_zf_lang1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => k2_zf_model(k4_zf_lang(A,B)) = k2_tarski(A,B) ) ) ), file(zf_lang1,t63_zf_lang1), []). fof(t64_zf_lang1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_zf_lang) => ! [B] : ( m2_subset_1(B,k5_numbers,k1_zf_lang) => k2_zf_model(k5_zf_lang(A,B)) = k2_tarski(A,B) ) ) ), file(zf_lang1,t64_zf_lang1), []). fof(t69_enumset1,axiom,( ! [A] : k2_tarski(A,A) = k1_tarski(A) ), file(enumset1,t69_enumset1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C,D] : ( ( v8_zf_fund1(B,A) & r2_hidden(C,B) & r2_hidden(D,B) ) => ( r2_hidden(k2_tarski(C,D),B) & r2_hidden(k4_tarski(C,D),B) ) ) ) ) ), file(zf_fund1,t6_zf_fund1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_zf_fund1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v8_zf_fund1(B,A) => r1_tarski(k5_ordinal2,B) ) ) ) ), file(zf_fund1,t7_zf_fund1), []). fof(t7_zf_model,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ! [B] : ( ~ v1_xboole_0(B) => ( v2_zf_lang(A) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,B) & m2_relset_1(C,k1_zf_lang,B) ) => ( k8_funct_2(k1_zf_lang,B,C,k18_zf_lang(A)) = k8_funct_2(k1_zf_lang,B,C,k19_zf_lang(A)) <=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ), file(zf_model,t7_zf_model), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_zf_model,axiom,( ! [A] : ( ( v1_zf_lang(A) & m2_finseq_1(A,k5_numbers) ) => ! [B] : ( ~ v1_xboole_0(B) => ( v3_zf_lang(A) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k1_zf_lang,B) & m2_relset_1(C,k1_zf_lang,B) ) => ( r2_hidden(k8_funct_2(k1_zf_lang,B,C,k18_zf_lang(A)),k8_funct_2(k1_zf_lang,B,C,k19_zf_lang(A))) <=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ), file(zf_model,t8_zf_model), []).