% Mizar problem: t61_card_fin,card_fin,2699,68 fof(t61_card_fin,conjecture,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_card_fin(B) ) => ! [C] : ( v1_finset_1(C) => ! [D,E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( k1_relat_1(B) = C & r2_hidden(D,k1_relat_1(B)) ) => ( r1_xreal_0(A,0) | k5_card_fin(k23_binop_2(A,1),B) = k23_binop_2(k5_card_fin(k23_binop_2(A,1),k7_relat_1(B,k4_xboole_0(k1_relat_1(B),k1_tarski(D)))),k5_card_fin(A,k1_yellow20(k7_relat_1(B,k4_xboole_0(k1_relat_1(B),k1_tarski(D))),k2_funcop_1(k4_xboole_0(k1_relat_1(B),k1_tarski(D)),k1_funct_1(B,D))))) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_afinsq_1,cc1_card_1,cc1_finseq_1,cc1_finset_1,cc1_fraenkel,cc1_funct_1,cc1_funct_2,cc1_membered,cc1_nat_1,cc1_ordinal2,cc1_relat_1,cc1_relset_1,cc1_stirl2_1,cc20_membered,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_membered,cc2_nat_1,cc3_card_1,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,commutativity_k1_nat_1,commutativity_k1_yellow20,commutativity_k23_binop_2,commutativity_k2_stirl2_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,connectedness_r1_xreal_0,d13_funct_1,d1_afinsq_1,d1_card_fin,d1_funct_2,d1_tarski,d1_xboole_0,d23_binop_2,d2_xboole_0,d3_xboole_0,d4_afinsq_1,d4_card_fin,d4_stirl2_1,d5_funct_1,d8_funct_1,dt_k10_binop_2,dt_k10_relat_1,dt_k10_stirl2_1,dt_k11_stirl2_1,dt_k1_afinsq_1,dt_k1_binop_1,dt_k1_card_1,dt_k1_card_fin,dt_k1_funct_1,dt_k1_funct_2,dt_k1_nat_1,dt_k1_numbers,dt_k1_ordinal4,dt_k1_relat_1,dt_k1_stirl2_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_yellow20,dt_k1_zfmisc_1,dt_k23_binop_2,dt_k2_afinsq_1,dt_k2_binop_1,dt_k2_card_fin,dt_k2_funcop_1,dt_k2_partfun1,dt_k2_relat_1,dt_k2_stirl2_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_funct_2,dt_k3_tarski,dt_k3_xboole_0,dt_k47_binop_2,dt_k4_card_1,dt_k4_relset_1,dt_k4_xboole_0,dt_k5_afinsq_1,dt_k5_card_fin,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relset_1,dt_k6_xcmplx_0,dt_k7_relat_1,dt_k7_stirl2_1,dt_m1_ordinal1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,existence_m1_ordinal1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_finset_1,fc10_membered,fc11_finseq_1,fc11_finset_1,fc11_membered,fc12_finset_1,fc12_membered,fc12_relat_1,fc13_membered,fc13_relat_1,fc14_finset_1,fc14_membered,fc15_binop_2,fc15_membered,fc16_membered,fc17_finseq_1,fc18_finseq_1,fc1_afinsq_1,fc1_card_fin,fc1_finset_1,fc1_fraenkel,fc1_funcop_1,fc1_funct_2,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc1_relat_1,fc1_stirl2_1,fc1_subset_1,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc27_membered,fc28_membered,fc29_membered,fc2_afinsq_1,fc2_card_1,fc2_card_fin,fc2_finseq_1,fc2_finset_1,fc2_fraenkel,fc2_funcop_1,fc2_funct_2,fc2_membered,fc2_ordinal2,fc2_relat_1,fc2_subset_1,fc2_xboole_0,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc37_membered,fc38_membered,fc39_membered,fc3_afinsq_1,fc3_funcop_1,fc3_nat_1,fc3_ordinal2,fc3_relat_1,fc3_subset_1,fc3_xboole_0,fc40_membered,fc41_membered,fc4_card_fin,fc4_funct_1,fc4_int_1,fc4_nat_1,fc4_ordinal2,fc4_relat_1,fc4_subset_1,fc5_membered,fc5_relat_1,fc6_funct_1,fc6_int_1,fc6_membered,fc6_relat_1,fc7_funcop_1,fc7_membered,fc7_relat_1,fc8_int_1,fc8_membered,fc8_relat_1,fc9_binop_2,fc9_finset_1,fc9_funcop_1,fc9_int_1,fc9_membered,fraenkel_a_3_0_card_fin,fraenkel_a_3_1_card_fin,fraenkel_a_5_4_card_fin,fraenkel_a_5_6_card_fin,fraenkel_a_5_8_card_fin,fraenkel_a_5_9_card_fin,idempotence_k2_xboole_0,idempotence_k3_xboole_0,l1_card_fin,l20_card_fin,l2_card_fin,l56_card_fin,rc1_afinsq_1,rc1_card_1,rc1_card_fin,rc1_finseq_1,rc1_finset_1,rc1_fraenkel,rc1_funcop_1,rc1_funct_1,rc1_funct_2,rc1_int_1,rc1_membered,rc1_nat_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_afinsq_1,rc2_card_1,rc2_card_fin,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_ordinal2,rc2_relat_1,rc2_subset_1,rc2_xboole_0,rc3_afinsq_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_nat_1,rc3_relat_1,rc4_afinsq_1,rc4_finset_1,rc4_funct_1,rc5_afinsq_1,rc5_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,redefinition_k10_binop_2,redefinition_k11_stirl2_1,redefinition_k1_afinsq_1,redefinition_k1_nat_1,redefinition_k1_stirl2_1,redefinition_k23_binop_2,redefinition_k2_afinsq_1,redefinition_k2_binop_1,redefinition_k2_partfun1,redefinition_k2_stirl2_1,redefinition_k3_funct_2,redefinition_k4_card_1,redefinition_k4_relset_1,redefinition_k5_afinsq_1,redefinition_k5_numbers,redefinition_k5_relset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_funct_2__e10_74__card_fin,s1_funct_2__e26_74__card_fin,spc0_boole,spc1_boole,spc1_numerals,symmetry_r1_xboole_0,t10_xreal_1,t13_card_fin,t13_finset_1,t13_xboole_1,t14_card_fin,t15_card_fin,t167_relat_1,t16_card_fin,t17_card_fin,t17_xboole_1,t1_arithm,t1_boole,t1_euler_1,t1_numerals,t1_real,t1_subset,t1_xboole_1,t28_xboole_1,t29_afinsq_1,t2_boole,t2_real,t2_subset,t2_tarski,t2_xreal_1,t30_card_fin,t32_card_fin,t34_card_fin,t36_xboole_1,t3_boole,t3_real,t3_subset,t47_card_1,t47_stirl2_1,t4_arithm,t4_boole,t4_funct_2,t4_real,t4_subset,t54_card_2,t57_card_fin,t59_card_fin,t5_real,t5_subset,t60_card_fin,t64_xboole_1,t64_zfmisc_1,t68_funct_1,t6_boole,t6_real,t71_funct_1,t7_afinsq_1,t7_boole,t7_real,t7_xboole_1,t87_relat_1,t8_boole,t8_real,t99_relat_1,t9_funct_1,t9_xreal_1]), [file(card_fin,t61_card_fin)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_afinsq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(A) ) ), file(afinsq_1,cc1_afinsq_1), []). 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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_fraenkel,axiom,( ! [A] : ( v1_fraenkel(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ) ), file(fraenkel,cc1_fraenkel), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_partfun1(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) ) ) ) ), file(funct_2,cc1_funct_2), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_nat_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) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_ordinal2,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_ordinal1(B,A) => v1_ordinal2(B) ) ) ), file(ordinal2,cc1_ordinal2), []). fof(cc1_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_relat_1(A) ) ), file(relat_1,cc1_relat_1), []). 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(cc1_stirl2_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ! [B] : ( m1_subset_1(B,A) => ( v4_ordinal2(B) & v1_xcmplx_0(B) & v1_finset_1(B) & v1_xreal_0(B) & ~ v3_xreal_0(B) & v1_int_1(B) ) ) ) ), file(stirl2_1,cc1_stirl2_1), []). fof(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). 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_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_int_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_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_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) ) ) ), file(nat_1,cc2_nat_1), []). 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_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc5_funct_2,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc5_funct_2), []). fof(cc6_funct_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & ~ v1_xboole_0(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc6_funct_2), []). fof(cc6_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). fof(cc9_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), []). 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_k1_yellow20,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => k1_yellow20(A,B) = k1_yellow20(B,A) ) ), file(yellow20,k1_yellow20), []). fof(commutativity_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k23_binop_2(B,A) ) ), file(binop_2,k23_binop_2), []). fof(commutativity_k2_stirl2_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_stirl2_1(A,B) = k2_stirl2_1(B,A) ) ), file(stirl2_1,k2_stirl2_1), []). 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_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_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). 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(d13_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k10_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,k1_relat_1(A)) & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ), file(funct_1,d13_funct_1), []). fof(d1_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k1_afinsq_1(A) <=> B = k1_relat_1(A) ) ) ) ), file(afinsq_1,d1_afinsq_1), []). fof(d1_card_fin,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C,D,E] : ( m1_subset_1(E,k1_zfmisc_1(k1_funct_2(A,k2_tarski(C,D)))) => ( E = k1_card_fin(A,B,C,D) <=> ! [F] : ( r2_hidden(F,E) <=> ? [G] : ( v1_funct_1(G) & v1_funct_2(G,A,k2_tarski(C,D)) & m2_relset_1(G,A,k2_tarski(C,D)) & G = F & k1_card_1(k3_funct_2(A,k2_tarski(C,D),G,k1_tarski(C))) = B ) ) ) ) ) ) ), file(card_fin,d1_card_fin), []). 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_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d23_binop_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) & m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) ) => ( A = k47_binop_2 <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_binop_1(k5_numbers,k5_numbers,k5_numbers,A,B,C) = k23_binop_2(B,C) ) ) ) ) ), file(binop_2,d23_binop_2), []). 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(d3_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d4_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v5_ordinal1(C) ) => ( C = k1_ordinal4(A,B) <=> ( k1_relat_1(C) = k1_nat_1(k1_afinsq_1(A),k1_afinsq_1(B)) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_afinsq_1(A)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_afinsq_1(B)) => k1_funct_1(C,k1_nat_1(k1_afinsq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ), file(afinsq_1,d4_afinsq_1), []). fof(d4_card_fin,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_card_fin(B) ) => ( v1_finset_1(k1_relat_1(B)) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( C = k5_card_fin(A,B) <=> ! [D,E,F] : ( v1_finset_1(F) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,k4_card_1(k1_card_fin(F,A,D,E)),k1_card_fin(F,A,D,E)) & m2_relset_1(G,k4_card_1(k1_card_fin(F,A,D,E)),k1_card_fin(F,A,D,E)) ) => ~ ( k1_relat_1(B) = F & v2_funct_1(G) & D != E & ! [H] : ( ( v1_finset_1(H) & m1_ordinal1(H,k5_numbers) ) => ~ ( k2_afinsq_1(H) = k4_relset_1(k4_card_1(k1_card_fin(F,A,D,E)),k1_card_fin(F,A,D,E),G) & ! [I,J] : ( ( v1_relat_1(J) & v1_funct_1(J) ) => ( ( r2_hidden(I,k2_afinsq_1(H)) & J = k1_funct_1(G,I) ) => k11_stirl2_1(H,I) = k1_card_1(k2_card_fin(B,J,D)) ) ) & C = k10_stirl2_1(H) ) ) ) ) ) ) ) ) ) ) ), file(card_fin,d4_card_fin), []). fof(d4_stirl2_1,axiom,( ! [A] : ( ( v1_finset_1(A) & m1_ordinal1(A,k5_numbers) ) => k10_stirl2_1(A) = k7_stirl2_1(k5_numbers,A,k47_binop_2) ) ), file(stirl2_1,d4_stirl2_1), []). 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(d8_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), []). fof(dt_k10_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k10_binop_2(A,B),k1_numbers) ) ), file(binop_2,k10_binop_2), []). fof(dt_k10_relat_1,axiom,( $true ), file(relat_1,k10_relat_1), []). fof(dt_k10_stirl2_1,axiom,( ! [A] : ( ( v1_finset_1(A) & m1_ordinal1(A,k5_numbers) ) => m2_subset_1(k10_stirl2_1(A),k1_numbers,k5_numbers) ) ), file(stirl2_1,k10_stirl2_1), []). fof(dt_k11_stirl2_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & m1_ordinal1(A,k5_numbers) ) => m2_subset_1(k11_stirl2_1(A,B),k1_numbers,k5_numbers) ) ), file(stirl2_1,k11_stirl2_1), []). fof(dt_k1_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => m2_subset_1(k1_afinsq_1(A),k1_numbers,k5_numbers) ) ), file(afinsq_1,k1_afinsq_1), []). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_card_fin,axiom,( ! [A,B,C,D] : ( ( v1_finset_1(A) & m1_subset_1(B,k5_numbers) ) => m1_subset_1(k1_card_fin(A,B,C,D),k1_zfmisc_1(k1_funct_2(A,k2_tarski(C,D)))) ) ), file(card_fin,k1_card_fin), []). 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_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_ordinal4,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) ) => ( v1_relat_1(k1_ordinal4(A,B)) & v1_funct_1(k1_ordinal4(A,B)) & v5_ordinal1(k1_ordinal4(A,B)) ) ) ), file(ordinal4,k1_ordinal4), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_stirl2_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k1_stirl2_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(stirl2_1,k1_stirl2_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_yellow20,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k1_yellow20(A,B)) & v1_funct_1(k1_yellow20(A,B)) ) ) ), file(yellow20,k1_yellow20), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k23_binop_2(A,B),k1_numbers,k5_numbers) ) ), file(binop_2,k23_binop_2), []). fof(dt_k2_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => m1_subset_1(k2_afinsq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(afinsq_1,k2_afinsq_1), []). fof(dt_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ), file(binop_1,k2_binop_1), []). fof(dt_k2_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => m1_subset_1(k2_card_fin(A,B,C),k1_zfmisc_1(k3_tarski(k2_relat_1(A)))) ) ), file(card_fin,k2_card_fin), []). fof(dt_k2_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), []). fof(dt_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => ( v1_funct_1(k2_partfun1(A,B,C,D)) & m2_relset_1(k2_partfun1(A,B,C,D),A,B) ) ) ), file(partfun1,k2_partfun1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_stirl2_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m1_subset_1(k2_stirl2_1(A,B),k1_zfmisc_1(k5_numbers)) ) ), file(stirl2_1,k2_stirl2_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_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k3_funct_2(A,B,C,D),k1_zfmisc_1(A)) ) ), file(funct_2,k3_funct_2), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k47_binop_2,axiom, ( v1_funct_1(k47_binop_2) & v1_funct_2(k47_binop_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) & m2_relset_1(k47_binop_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) ), file(binop_2,k47_binop_2), []). fof(dt_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) ), file(card_1,k4_card_1), []). 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_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), []). fof(dt_k5_afinsq_1,axiom,( ! [A,B,C] : ( ( v1_finset_1(B) & m1_ordinal1(B,A) & v1_finset_1(C) & m1_ordinal1(C,A) ) => m1_ordinal1(k5_afinsq_1(A,B,C),A) ) ), file(afinsq_1,k5_afinsq_1), []). fof(dt_k5_card_fin,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_relat_1(B) & v1_funct_1(B) & v1_card_fin(B) ) => m2_subset_1(k5_card_fin(A,B),k1_numbers,k5_numbers) ) ), file(card_fin,k5_card_fin), []). 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_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ), file(relset_1,k5_relset_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). 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_stirl2_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_finset_1(B) & m1_ordinal1(B,A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) ) => m1_subset_1(k7_stirl2_1(A,B,C),A) ) ), file(stirl2_1,k7_stirl2_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_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_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_ordinal1,axiom,( ! [A] : ? [B] : m1_ordinal1(B,A) ), file(ordinal1,m1_ordinal1), []). 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_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(fc10_finset_1,axiom,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_1), []). fof(fc10_membered,axiom,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), []). fof(fc11_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k2_relat_1(A)) ) ), file(finseq_1,fc11_finseq_1), []). fof(fc11_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), []). fof(fc11_membered,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), []). 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(fc12_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc12_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) ), file(relat_1,fc12_relat_1), []). fof(fc13_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(fc13_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v3_relat_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v3_relat_1(k7_relat_1(A,B)) ) ) ), file(relat_1,fc13_relat_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(fc14_membered,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), []). fof(fc15_binop_2,axiom, ( v1_funct_1(k47_binop_2) & v1_funct_2(k47_binop_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) & v1_binop_1(k47_binop_2,k5_numbers) & v2_binop_1(k47_binop_2,k5_numbers) & v1_setwiseo(k47_binop_2,k5_numbers) ), file(binop_2,fc15_binop_2), []). fof(fc15_membered,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), []). fof(fc16_membered,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), []). fof(fc17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_1), []). fof(fc18_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k10_relat_1(A,B)) ) ), file(finseq_1,fc18_finseq_1), []). fof(fc1_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => ( v1_ordinal1(k1_relat_1(A)) & v2_ordinal1(k1_relat_1(A)) & v3_ordinal1(k1_relat_1(A)) & v4_ordinal2(k1_relat_1(A)) & v1_xcmplx_0(k1_relat_1(A)) & v1_xreal_0(k1_relat_1(A)) ) ) ), file(afinsq_1,fc1_afinsq_1), []). fof(fc1_card_fin,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_card_fin(A) ) => v1_finset_1(k1_funct_1(A,B)) ) ), file(card_fin,fc1_card_fin), []). 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_fraenkel,axiom,( ! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ), file(fraenkel,fc1_fraenkel), []). fof(fc1_funcop_1,axiom,( ! [A,B] : ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) ) ), file(funcop_1,fc1_funcop_1), []). fof(fc1_funct_2,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ~ v1_xboole_0(k1_funct_2(A,B)) ) ), file(funct_2,fc1_funct_2), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_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_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k3_xboole_0(A,B)) ) ), file(relat_1,fc1_relat_1), []). fof(fc1_stirl2_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(B,k1_zfmisc_1(k5_numbers)) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => ( v4_ordinal2(k1_funct_1(C,D)) & v1_xcmplx_0(k1_funct_1(C,D)) & v1_finset_1(k1_funct_1(C,D)) & v1_xreal_0(k1_funct_1(C,D)) & ~ v3_xreal_0(k1_funct_1(C,D)) & v1_int_1(k1_funct_1(C,D)) ) ) ), file(stirl2_1,fc1_stirl2_1), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc23_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), []). fof(fc24_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), []). fof(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc26_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), []). fof(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). fof(fc29_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), []). fof(fc2_afinsq_1,axiom, ( v1_xboole_0(k1_xboole_0) & 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_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) & v5_ordinal1(k1_xboole_0) & v4_ordinal2(k1_xboole_0) & v1_xcmplx_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_funct_7(k1_xboole_0) & v1_xreal_0(k1_xboole_0) & ~ v2_xreal_0(k1_xboole_0) & ~ v3_xreal_0(k1_xboole_0) ), file(afinsq_1,fc2_afinsq_1), []). 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_card_fin,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_card_fin(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) & v1_card_fin(k7_relat_1(A,B)) ) ) ), file(card_fin,fc2_card_fin), []). fof(fc2_finseq_1,axiom, ( v1_xboole_0(k1_xboole_0) & 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_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), []). 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_fraenkel,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => ( v1_finset_1(k1_funct_2(A,B)) & v1_fraenkel(k1_funct_2(A,B)) ) ) ), file(fraenkel,fc2_fraenkel), []). fof(fc2_funcop_1,axiom,( ! [A] : ( v1_xboole_0(k2_funcop_1(k1_xboole_0,A)) & v1_relat_1(k2_funcop_1(k1_xboole_0,A)) & v1_funct_1(k2_funcop_1(k1_xboole_0,A)) & v2_funct_1(k2_funcop_1(k1_xboole_0,A)) ) ), file(funcop_1,fc2_funcop_1), []). fof(fc2_funct_2,axiom,( ! [A] : ~ v1_xboole_0(k1_funct_2(A,A)) ), file(funct_2,fc2_funct_2), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). 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_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k2_xboole_0(A,B)) ) ), file(relat_1,fc2_relat_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ), file(xboole_0,fc2_xboole_0), []). fof(fc30_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), []). fof(fc31_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), []). fof(fc32_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), []). fof(fc33_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), []). fof(fc34_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), []). fof(fc35_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), []). fof(fc36_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), []). fof(fc37_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k4_xboole_0(A,B)) ) ), file(membered,fc37_membered), []). fof(fc38_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc38_membered), []). fof(fc39_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc39_membered), []). fof(fc3_afinsq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) & v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) => ( v1_relat_1(k1_ordinal4(A,B)) & v1_funct_1(k1_ordinal4(A,B)) & v5_ordinal1(k1_ordinal4(A,B)) & v1_finset_1(k1_ordinal4(A,B)) ) ) ), file(afinsq_1,fc3_afinsq_1), []). fof(fc3_funcop_1,axiom,( ! [A,B] : ( v1_xboole_0(B) => ( v1_xboole_0(k2_funcop_1(B,A)) & v1_relat_1(k2_funcop_1(B,A)) & v1_funct_1(k2_funcop_1(B,A)) & v2_funct_1(k2_funcop_1(B,A)) ) ) ), file(funcop_1,fc3_funcop_1), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_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_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k4_xboole_0(A,B)) ) ), file(relat_1,fc3_relat_1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ), file(xboole_0,fc3_xboole_0), []). fof(fc40_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc40_membered), []). fof(fc41_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) & v5_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc41_membered), []). fof(fc4_card_fin,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_card_fin(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k1_yellow20(A,B)) & v1_funct_1(k1_yellow20(A,B)) & v1_card_fin(k1_yellow20(A,B)) ) ) ), file(card_fin,fc4_card_fin), []). 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_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), []). fof(fc4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). 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_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) ), file(relat_1,fc4_relat_1), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_relat_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_relat_1(A) ) => ~ v1_xboole_0(k1_relat_1(A)) ) ), file(relat_1,fc5_relat_1), []). 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_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc6_relat_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_relat_1(A) ) => ~ v1_xboole_0(k2_relat_1(A)) ) ), file(relat_1,fc6_relat_1), []). fof(fc7_funcop_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_funcop_1(A) ) => ( v1_relat_1(k1_funct_1(A,B)) & v1_funct_1(k1_funct_1(A,B)) ) ) ), file(funcop_1,fc7_funcop_1), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc7_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_xboole_0(k1_relat_1(A)) & v1_relat_1(k1_relat_1(A)) ) ) ), file(relat_1,fc7_relat_1), []). fof(fc8_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc8_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_xboole_0(k2_relat_1(A)) & v1_relat_1(k2_relat_1(A)) ) ) ), file(relat_1,fc8_relat_1), []). fof(fc9_binop_2,axiom, ( v1_funct_1(k47_binop_2) & v1_funct_2(k47_binop_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) & v1_binop_1(k47_binop_2,k5_numbers) & v2_binop_1(k47_binop_2,k5_numbers) ), file(binop_2,fc9_binop_2), []). 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(fc9_funcop_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ( v1_relat_1(k2_funcop_1(A,B)) & v2_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) ) ) ), file(funcop_1,fc9_funcop_1), []). fof(fc9_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), []). fof(fc9_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(fraenkel_a_3_0_card_fin,axiom,( ! [A,B,C,D] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & v1_finset_1(C) ) => ( r2_hidden(A,a_3_0_card_fin(B,C,D)) <=> ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k2_xboole_0(k4_xboole_0(C,k1_tarski(D)),k1_tarski(D)),k2_stirl2_1(1,0)) & m2_relset_1(E,k2_xboole_0(k4_xboole_0(C,k1_tarski(D)),k1_tarski(D)),k2_stirl2_1(1,0)) & A = E & k1_card_1(k3_funct_2(k2_xboole_0(k4_xboole_0(C,k1_tarski(D)),k1_tarski(D)),k2_stirl2_1(1,0),E,k1_stirl2_1(1))) = k23_binop_2(B,1) & k1_funct_1(E,D) = 0 ) ) ) ), file(card_fin,a_3_0_card_fin), []). fof(fraenkel_a_3_1_card_fin,axiom,( ! [A,B,C,D] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & v1_finset_1(C) ) => ( r2_hidden(A,a_3_1_card_fin(B,C,D)) <=> ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k2_xboole_0(k4_xboole_0(C,k1_tarski(D)),k1_tarski(D)),k2_stirl2_1(1,0)) & m2_relset_1(E,k2_xboole_0(k4_xboole_0(C,k1_tarski(D)),k1_tarski(D)),k2_stirl2_1(1,0)) & A = E & k1_card_1(k3_funct_2(k2_xboole_0(k4_xboole_0(C,k1_tarski(D)),k1_tarski(D)),k2_stirl2_1(1,0),E,k1_stirl2_1(1))) = k23_binop_2(B,1) & k1_funct_1(E,D) = 1 ) ) ) ), file(card_fin,a_3_1_card_fin), []). fof(fraenkel_a_5_4_card_fin,axiom,( ! [A,B,C,D,E,F] : ( ( v1_finset_1(E) & m2_subset_1(F,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_5_4_card_fin(B,C,D,E,F)) <=> ? [G] : ( v1_funct_1(G) & v1_funct_2(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D)) & m2_relset_1(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D)) & A = G & k1_card_1(k3_funct_2(k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D),G,k1_tarski(C))) = k23_binop_2(F,1) & k1_funct_1(G,B) = C ) ) ) ), file(card_fin,a_5_4_card_fin), []). fof(fraenkel_a_5_6_card_fin,axiom,( ! [A,B,C,D,E,F] : ( ( v1_finset_1(E) & m2_subset_1(F,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_5_6_card_fin(B,C,D,E,F)) <=> ? [G] : ( v1_funct_1(G) & v1_funct_2(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D)) & m2_relset_1(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D)) & A = G & k1_card_1(k3_funct_2(k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D),G,k1_tarski(C))) = F & k1_funct_1(G,B) = D ) ) ) ), file(card_fin,a_5_6_card_fin), []). fof(fraenkel_a_5_8_card_fin,axiom,( ! [A,B,C,D,E,F] : ( ( v1_finset_1(E) & m2_subset_1(F,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_5_8_card_fin(B,C,D,E,F)) <=> ? [G] : ( v1_funct_1(G) & v1_funct_2(G,k2_xboole_0(E,k1_tarski(D)),k2_tarski(B,C)) & m2_relset_1(G,k2_xboole_0(E,k1_tarski(D)),k2_tarski(B,C)) & A = G & k1_card_1(k3_funct_2(k2_xboole_0(E,k1_tarski(D)),k2_tarski(B,C),G,k1_tarski(B))) = k23_binop_2(F,1) & k1_funct_1(G,D) = B ) ) ) ), file(card_fin,a_5_8_card_fin), []). fof(fraenkel_a_5_9_card_fin,axiom,( ! [A,B,C,D,E,F] : ( ( v1_finset_1(E) & m2_subset_1(F,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_5_9_card_fin(B,C,D,E,F)) <=> ? [G] : ( v1_funct_1(G) & v1_funct_2(G,k2_xboole_0(E,k1_tarski(D)),k2_tarski(B,C)) & m2_relset_1(G,k2_xboole_0(E,k1_tarski(D)),k2_tarski(B,C)) & A = G & k1_card_1(k3_funct_2(k2_xboole_0(E,k1_tarski(D)),k2_tarski(B,C),G,k1_tarski(B))) = k23_binop_2(F,1) & k1_funct_1(G,D) = C ) ) ) ), file(card_fin,a_5_9_card_fin), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(l1_card_fin,axiom,( ! [A,B] : ( r2_hidden(A,B) => k2_xboole_0(k4_xboole_0(B,k1_tarski(A)),k1_tarski(A)) = B ) ), file(card_fin,l1_card_fin), []). fof(l20_card_fin,axiom,( ! [A,B,C,D] : ( v1_finset_1(D) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( A != B & ~ r2_hidden(C,D) & ~ ( k2_xboole_0(a_5_8_card_fin(A,B,C,D,E),a_5_9_card_fin(A,B,C,D,E)) = k1_card_fin(k2_xboole_0(D,k1_tarski(C)),k23_binop_2(E,1),A,B) & r1_xboole_0(a_5_8_card_fin(A,B,C,D,E),a_5_9_card_fin(A,B,C,D,E)) ) ) ) ) ), file(card_fin,l20_card_fin), []). fof(l2_card_fin,axiom,( ! [A,B] : ( ~ r2_hidden(A,B) => k4_xboole_0(k2_xboole_0(B,k1_tarski(A)),k1_tarski(A)) = B ) ), file(card_fin,l2_card_fin), []). fof(l56_card_fin,axiom,( ! [A] : ( v1_finset_1(A) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k4_card_1(A),A) & m2_relset_1(B,k4_card_1(A),A) & v2_funct_1(B) ) ) ), file(card_fin,l56_card_fin), []). fof(rc1_afinsq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) ), file(afinsq_1,rc1_afinsq_1), []). fof(rc1_card_1,axiom,( ? [A] : v1_card_1(A) ), file(card_1,rc1_card_1), []). fof(rc1_card_fin,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_card_fin(A) ) ), file(card_fin,rc1_card_fin), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_fraenkel,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_fraenkel(A) ) ), file(fraenkel,rc1_fraenkel), []). fof(rc1_funcop_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_funcop_1(A) ) ), file(funcop_1,rc1_funcop_1), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_funct_2,axiom,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,A,B) ) ), file(funct_2,rc1_funct_2), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_relat_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc1_relat_1), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_afinsq_1,axiom,( ! [A] : ? [B] : ( m1_ordinal1(B,A) & v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) ), file(afinsq_1,rc2_afinsq_1), []). 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_card_fin,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) & v1_finset_1(A) & v1_card_fin(A) ) ), file(card_fin,rc2_card_fin), []). fof(rc2_finseq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), []). 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_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_ordinal2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_ordinal2(A) ) ), file(ordinal2,rc2_ordinal2), []). fof(rc2_relat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc2_relat_1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_afinsq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) ), file(afinsq_1,rc3_afinsq_1), []). fof(rc3_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), []). 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_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_relat_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) ) ), file(relat_1,rc3_relat_1), []). fof(rc4_afinsq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v5_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_finset_1(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(afinsq_1,rc4_afinsq_1), []). 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(rc5_afinsq_1,axiom,( ! [A] : ? [B] : ( m1_ordinal1(B,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) & v5_ordinal1(B) & v4_ordinal2(B) & v1_xcmplx_0(B) & v1_finset_1(B) & v1_xreal_0(B) & ~ v2_xreal_0(B) & ~ v3_xreal_0(B) ) ), file(afinsq_1,rc5_afinsq_1), []). 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(rc6_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(rc7_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(redefinition_k10_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k10_binop_2(A,B) = k6_xcmplx_0(A,B) ) ), file(binop_2,k10_binop_2), []). fof(redefinition_k11_stirl2_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & m1_ordinal1(A,k5_numbers) ) => k11_stirl2_1(A,B) = k1_funct_1(A,B) ) ), file(stirl2_1,k11_stirl2_1), []). fof(redefinition_k1_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => k1_afinsq_1(A) = k1_card_1(A) ) ), file(afinsq_1,k1_afinsq_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_k1_stirl2_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => k1_stirl2_1(A) = k1_tarski(A) ) ), file(stirl2_1,k1_stirl2_1), []). fof(redefinition_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k23_binop_2), []). fof(redefinition_k2_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => k2_afinsq_1(A) = k1_relat_1(A) ) ), file(afinsq_1,k2_afinsq_1), []). fof(redefinition_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ), file(binop_1,k2_binop_1), []). fof(redefinition_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_partfun1(A,B,C,D) = k7_relat_1(C,D) ) ), file(partfun1,k2_partfun1), []). fof(redefinition_k2_stirl2_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_stirl2_1(A,B) = k2_tarski(A,B) ) ), file(stirl2_1,k2_stirl2_1), []). fof(redefinition_k3_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k3_funct_2(A,B,C,D) = k10_relat_1(C,D) ) ), file(funct_2,k3_funct_2), []). fof(redefinition_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_1), []). 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_k5_afinsq_1,axiom,( ! [A,B,C] : ( ( v1_finset_1(B) & m1_ordinal1(B,A) & v1_finset_1(C) & m1_ordinal1(C,A) ) => k5_afinsq_1(A,B,C) = k1_ordinal4(B,C) ) ), file(afinsq_1,k5_afinsq_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k5_relset_1(A,B,C) = k2_relat_1(C) ) ), file(relset_1,k5_relset_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(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(s1_funct_2__e10_74__card_fin,axiom,( ! [A,B,C,D,E] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & v1_finset_1(B) & v1_finset_1(D) & v1_funct_1(E) & v1_funct_2(E,k4_card_1(D),D) & m2_relset_1(E,k4_card_1(D),D) ) => ( ! [F] : ~ ( r2_hidden(F,k4_card_1(D)) & ! [G] : ~ ( r2_hidden(G,k1_card_fin(k4_xboole_0(B,k1_tarski(C)),k23_binop_2(A,1),1,0)) & ? [H] : ( v1_relat_1(H) & v1_funct_1(H) & H = k1_funct_1(E,F) & r2_hidden(H,D) & G = k7_relat_1(H,k4_xboole_0(B,k1_tarski(C))) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k4_card_1(D),k1_card_fin(k4_xboole_0(B,k1_tarski(C)),k23_binop_2(A,1),1,0)) & m2_relset_1(F,k4_card_1(D),k1_card_fin(k4_xboole_0(B,k1_tarski(C)),k23_binop_2(A,1),1,0)) & ! [G] : ( r2_hidden(G,k4_card_1(D)) => ? [I] : ( v1_relat_1(I) & v1_funct_1(I) & I = k1_funct_1(E,G) & r2_hidden(I,D) & k1_funct_1(F,G) = k7_relat_1(I,k4_xboole_0(B,k1_tarski(C))) ) ) ) ) ) ), file(card_fin,s1_funct_2__e10_74__card_fin), []). fof(s1_funct_2__e26_74__card_fin,axiom,( ! [A,B,C,D,E] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & v1_finset_1(B) & v1_finset_1(D) & v1_funct_1(E) & v1_funct_2(E,k4_card_1(D),D) & m2_relset_1(E,k4_card_1(D),D) ) => ( ! [F] : ~ ( r2_hidden(F,k4_card_1(D)) & ! [G] : ~ ( r2_hidden(G,k1_card_fin(k4_xboole_0(B,k1_tarski(C)),A,1,0)) & ? [H] : ( v1_relat_1(H) & v1_funct_1(H) & H = k1_funct_1(E,F) & r2_hidden(H,D) & G = k7_relat_1(H,k4_xboole_0(B,k1_tarski(C))) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k4_card_1(D),k1_card_fin(k4_xboole_0(B,k1_tarski(C)),A,1,0)) & m2_relset_1(F,k4_card_1(D),k1_card_fin(k4_xboole_0(B,k1_tarski(C)),A,1,0)) & ! [G] : ( r2_hidden(G,k4_card_1(D)) => ? [I] : ( v1_relat_1(I) & v1_funct_1(I) & I = k1_funct_1(E,G) & r2_hidden(I,D) & k1_funct_1(F,G) = k7_relat_1(I,k4_xboole_0(B,k1_tarski(C))) ) ) ) ) ) ), file(card_fin,s1_funct_2__e26_74__card_fin), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(t10_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(B,A) & r1_xreal_0(C,D) & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ), file(xreal_1,t10_xreal_1), []). fof(t13_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( k1_funct_1(C,A) = B & r2_hidden(A,k1_relat_1(C)) ) => k2_xboole_0(k1_tarski(A),k10_relat_1(k7_relat_1(C,k4_xboole_0(k1_relat_1(C),k1_tarski(A))),k1_tarski(B))) = k10_relat_1(C,k1_tarski(B)) ) ) ), file(card_fin,t13_card_fin), []). fof(t13_finset_1,axiom,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), []). fof(t13_xboole_1,axiom,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) ) => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) ), file(xboole_1,t13_xboole_1), []). fof(t14_card_fin,axiom,( ! [A,B,C,D] : ( v1_finset_1(D) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ~ r2_hidden(A,D) => k4_card_1(k1_card_fin(D,E,B,C)) = k1_card_1(a_5_4_card_fin(A,B,C,D,E)) ) ) ) ), file(card_fin,t14_card_fin), []). fof(t15_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( k1_funct_1(C,A) != B => k10_relat_1(k7_relat_1(C,k4_xboole_0(k1_relat_1(C),k1_tarski(A))),k1_tarski(B)) = k10_relat_1(C,k1_tarski(B)) ) ) ), file(card_fin,t15_card_fin), []). fof(t167_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k10_relat_1(B,A),k1_relat_1(B)) ) ), file(relat_1,t167_relat_1), []). fof(t16_card_fin,axiom,( ! [A,B,C,D] : ( v1_finset_1(D) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( ~ r2_hidden(A,D) & B != C & k4_card_1(k1_card_fin(D,E,B,C)) != k1_card_1(a_5_6_card_fin(A,B,C,D,E)) ) ) ) ), file(card_fin,t16_card_fin), []). fof(t17_card_fin,axiom,( ! [A,B,C,D] : ( v1_finset_1(D) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( A != B & ~ r2_hidden(C,D) & k4_card_1(k1_card_fin(k2_xboole_0(D,k1_tarski(C)),k23_binop_2(E,1),A,B)) != k23_binop_2(k4_card_1(k1_card_fin(D,k23_binop_2(E,1),A,B)),k4_card_1(k1_card_fin(D,E,A,B))) ) ) ) ), file(card_fin,t17_card_fin), []). fof(t17_xboole_1,axiom,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_xboole_1), []). 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_euler_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,B) <=> ~ r1_xreal_0(B,A) ) ) ) ), file(euler_1,t1_euler_1), []). 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(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t29_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) => k2_relat_1(k1_ordinal4(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ), file(afinsq_1,t29_afinsq_1), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). 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(t2_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(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), []). fof(t30_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) ) => ( k10_relat_1(C,k1_tarski(A)) = k10_relat_1(k7_relat_1(C,B),k1_tarski(A)) => k2_card_fin(D,C,A) = k2_card_fin(D,k7_relat_1(C,B),A) ) ) ) ), file(card_fin,t30_card_fin), []). fof(t32_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) ) => ( ( r2_hidden(A,k2_relat_1(C)) & r1_tarski(k10_relat_1(C,k1_tarski(A)),B) ) => k2_card_fin(k7_relat_1(D,B),C,A) = k2_card_fin(D,C,A) ) ) ) ), file(card_fin,t32_card_fin), []). fof(t34_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) ) => ( r2_hidden(A,k10_relat_1(C,k1_tarski(B))) => k3_xboole_0(k2_card_fin(D,k7_relat_1(C,k4_xboole_0(k1_relat_1(C),k1_tarski(A))),B),k1_funct_1(D,A)) = k2_card_fin(D,C,B) ) ) ) ), file(card_fin,t34_card_fin), []). fof(t36_xboole_1,axiom,( ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ), file(xboole_1,t36_xboole_1), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). 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(t47_card_1,axiom,( k1_card_1(k1_xboole_0) = k1_xboole_0 ), file(card_1,t47_card_1), []). fof(t47_stirl2_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_finset_1(B) & m1_ordinal1(B,A) ) => ! [C] : ( ( v1_finset_1(C) & m1_ordinal1(C,A) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,A),A) & m2_relset_1(D,k2_zfmisc_1(A,A),A) ) => ( v2_binop_1(D,A) => ( ( ~ v1_setwiseo(D,A) & ~ ( r1_xreal_0(1,k1_afinsq_1(B)) & r1_xreal_0(1,k1_afinsq_1(C)) ) ) | k7_stirl2_1(A,k5_afinsq_1(A,B,C),D) = k2_binop_1(A,A,A,D,k7_stirl2_1(A,B,D),k7_stirl2_1(A,C,D)) ) ) ) ) ) ) ), file(stirl2_1,t47_stirl2_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). fof(t4_funct_2,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(k2_relat_1(B),A) => ( v1_funct_1(B) & v1_funct_2(B,k1_relat_1(B),A) & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ), file(funct_2,t4_funct_2), []). 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(t54_card_2,axiom,( ! [A,B] : ( v1_finset_1(B) => ( ~ r2_hidden(A,B) => k4_card_1(k2_xboole_0(B,k1_tarski(A))) = k1_nat_1(k4_card_1(B),1) ) ) ), file(card_2,t54_card_2), []). fof(t57_card_fin,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( k1_relat_1(k1_yellow20(B,k2_funcop_1(k1_relat_1(B),A))) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(B)) => k1_funct_1(k1_yellow20(B,k2_funcop_1(k1_relat_1(B),A)),C) = k3_xboole_0(k1_funct_1(B,C),A) ) ) ) ), file(card_fin,t57_card_fin), []). fof(t59_card_fin,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) ) => k3_xboole_0(k2_card_fin(C,D,A),B) = k2_card_fin(k1_yellow20(C,k2_funcop_1(k1_relat_1(C),B)),D,A) ) ) ), file(card_fin,t59_card_fin), []). 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(t60_card_fin,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) => ( ( v2_funct_1(A) & v2_funct_1(B) & r1_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) => v2_funct_1(k1_ordinal4(A,B)) ) ) ) ), file(card_fin,t60_card_fin), []). fof(t64_xboole_1,axiom,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) & r1_xboole_0(B,D) ) => r1_xboole_0(A,C) ) ), file(xboole_1,t64_xboole_1), []). fof(t64_zfmisc_1,axiom,( ! [A,B,C] : ( r2_hidden(A,k4_xboole_0(B,k1_tarski(C))) <=> ( r2_hidden(A,B) & A != C ) ) ), file(zfmisc_1,t64_zfmisc_1), []). fof(t68_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( B = k7_relat_1(C,A) <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A) & ! [D] : ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ), file(funct_1,t68_funct_1), []). 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(t71_funct_1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,k3_xboole_0(k1_relat_1(C),A)) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t71_funct_1), []). fof(t7_afinsq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( ( v1_finset_1(A) & v5_ordinal1(A) ) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & k1_relat_1(A) = B ) ) ) ), file(afinsq_1,t7_afinsq_1), []). 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(t87_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k1_relat_1(k7_relat_1(B,A)),A) ) ), file(relat_1,t87_relat_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_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(t99_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k2_relat_1(k7_relat_1(B,A)),k2_relat_1(B)) ) ), file(relat_1,t99_relat_1), []). fof(t9_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) => A = B ) ) ) ), file(funct_1,t9_funct_1), []). 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), []).