% Mizar problem: t34_modal_1,modal_1,1582,11 fof(t34_modal_1,conjecture,( ! [A] : ( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) => ! [B] : ( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) => m2_modal_1(k8_trees_2(k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(2),k1_domain_1(k5_numbers,k5_numbers,2,0)),k12_finseq_1(k5_numbers,0),A),k12_finseq_1(k5_numbers,1),B),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,antisymmetry_r2_xboole_0,cc1_arytm_3,cc1_finseq_1,cc1_finset_1,cc1_modal_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,commutativity_k2_tarski,commutativity_k7_domain_1,connectedness_r1_xreal_0,d12_trees_1,d12_trees_2,d13_trees_2,d1_enumset1,d2_tarski,d5_tarski,d5_trees_2,d6_modal_1,d8_modal_1,d8_xboole_0,dt_k10_trees_2,dt_k12_finseq_1,dt_k12_mcart_1,dt_k1_card_1,dt_k1_domain_1,dt_k1_enumset1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_trees_2,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_relat_1,dt_k2_tarski,dt_k2_trees_1,dt_k2_zfmisc_1,dt_k3_trees_2,dt_k4_card_1,dt_k4_tarski,dt_k5_finseq_1,dt_k5_modal_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_trees_1,dt_k6_modal_1,dt_k7_domain_1,dt_k7_finseq_1,dt_k8_finseq_1,dt_k8_trees_2,dt_k9_finseq_1,dt_k9_modal_1,dt_k9_trees_2,dt_m1_finseq_1,dt_m1_modal_1,dt_m1_relset_1,dt_m1_subset_1,dt_m1_trees_1,dt_m2_finseq_1,dt_m2_modal_1,dt_m2_relset_1,dt_m2_subset_1,dt_m3_trees_2,existence_m1_finseq_1,existence_m1_modal_1,existence_m1_relset_1,existence_m1_subset_1,existence_m1_trees_1,existence_m2_finseq_1,existence_m2_modal_1,existence_m2_relset_1,existence_m2_subset_1,existence_m3_trees_2,fc11_finseq_1,fc12_finseq_1,fc13_finseq_1,fc14_finseq_1,fc14_finset_1,fc17_finseq_1,fc1_finset_1,fc1_ordinal2,fc1_subset_1,fc2_finseq_1,fc2_finset_1,fc2_subset_1,fc2_trees_1,fc2_trees_2,fc3_finseq_1,fc3_finset_1,fc3_subset_1,fc3_trees_2,fc4_finseq_1,fc4_ordinal2,fc4_subset_1,fc4_trees_2,fc5_trees_2,fraenkel_a_1_3_modal_1,fraenkel_a_2_1_trees_2,irreflexivity_r2_xboole_0,l1_modal_1,l31_modal_1,rc1_arytm_3,rc1_finseq_1,rc1_finset_1,rc1_subset_1,rc1_trees_1,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_subset_1,rc2_trees_1,rc2_xreal_0,rc3_finseq_1,rc3_finset_1,rc3_xreal_0,rc4_finseq_1,rc4_finset_1,rc4_xreal_0,rc5_trees_2,rc6_finseq_1,rc6_trees_2,rc7_finseq_1,rc7_trees_2,rc8_finseq_1,rc8_trees_2,rc9_trees_2,redefinition_k12_finseq_1,redefinition_k12_mcart_1,redefinition_k1_domain_1,redefinition_k3_trees_2,redefinition_k4_card_1,redefinition_k5_modal_1,redefinition_k5_numbers,redefinition_k7_domain_1,redefinition_k8_finseq_1,redefinition_k9_finseq_1,redefinition_k9_trees_2,redefinition_m1_trees_1,redefinition_m2_finseq_1,redefinition_m2_modal_1,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r2_wellord2,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r2_wellord2,reflexivity_r3_xboole_0,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,symmetry_r2_wellord2,symmetry_r3_xboole_0,t10_modal_1,t13_funcop_1,t16_trees_1,t17_modal_1,t18_modal_1,t19_funcop_1,t19_modal_1,t1_numerals,t1_real,t1_subset,t2_real,t2_subset,t2_tarski,t3_modal_1,t3_real,t3_subset,t47_finseq_1,t4_real,t4_subset,t55_trees_1,t5_real,t5_subset,t6_boole,t6_real,t76_card_2,t7_boole,t7_real,t81_card_1,t8_boole,t8_real,t8_trees_1]), [file(modal_1,t34_modal_1)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(antisymmetry_r2_xboole_0,axiom,( ! [A,B] : ( r2_xboole_0(A,B) => ~ r2_xboole_0(B,A) ) ), file(xboole_0,r2_xboole_0), []). fof(cc1_arytm_3,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), []). 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_modal_1,axiom,( ! [A] : ( m1_subset_1(A,k6_modal_1) => v1_finset_1(A) ) ), file(modal_1,cc1_modal_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_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc2_arytm_3,axiom,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), []). 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_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_arytm_3,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), []). fof(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). fof(cc4_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), []). fof(cc5_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), []). fof(cc6_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), []). fof(cc7_xreal_0,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k7_domain_1(A,C,B) ) ), file(domain_1,k7_domain_1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d12_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ! [C] : ( ( ~ v1_xboole_0(C) & v1_trees_1(C) ) => ( r2_hidden(B,A) => ! [D] : ( ( ~ v1_xboole_0(D) & v1_trees_1(D) ) => ( D = k5_trees_1(A,B,C) <=> ! [E] : ( m2_finseq_1(E,k5_numbers) => ( r2_hidden(E,D) <=> ~ ( ~ ( r2_hidden(E,A) & ~ r2_xboole_0(B,E) ) & ! [F] : ( m2_finseq_1(F,k5_numbers) => ~ ( r2_hidden(F,C) & E = k8_finseq_1(k5_numbers,B,F) ) ) ) ) ) ) ) ) ) ) ) ), file(trees_1,d12_trees_1), []). fof(d12_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( r2_hidden(B,k1_relat_1(A)) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v3_trees_2(D) ) => ( D = k8_trees_2(A,B,C) <=> ( k1_relat_1(D) = k5_trees_1(k1_relat_1(A),B,k1_relat_1(C)) & ! [E] : ( m2_finseq_1(E,k5_numbers) => ~ ( r2_hidden(E,k5_trees_1(k1_relat_1(A),B,k1_relat_1(C))) & ~ ( ~ r1_tarski(B,E) & k1_funct_1(D,E) = k1_funct_1(A,E) ) & ! [F] : ( m2_finseq_1(F,k5_numbers) => ~ ( r2_hidden(F,k1_relat_1(C)) & E = k8_finseq_1(k5_numbers,B,F) & k1_funct_1(D,E) = k1_funct_1(C,F) ) ) ) ) ) ) ) ) ) ) ) ), file(trees_2,d12_trees_2), []). fof(d13_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) => k10_trees_2(A,B) = k4_card_1(k1_trees_2(A,B)) ) ) ), file(trees_2,d13_trees_2), []). fof(d1_enumset1,axiom,( ! [A,B,C,D] : ( D = k1_enumset1(A,B,C) <=> ! [E] : ( r2_hidden(E,D) <=> ~ ( E != A & E != B & E != C ) ) ) ), file(enumset1,d1_enumset1), []). fof(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(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d5_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) => k1_trees_2(A,B) = a_2_1_trees_2(A,B) ) ) ), file(trees_2,d5_trees_2), []). fof(d6_modal_1,axiom,( ! [A] : ( m1_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) => ( A = k6_modal_1 <=> ( ! [B] : ( ( v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ) => ( r2_hidden(B,A) => v1_finset_1(B) ) ) & ! [B] : ( ( v1_funct_1(B) & v1_finset_1(B) & v3_trees_2(B) & m3_trees_2(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ) => ( r2_hidden(B,A) <=> ! [C] : ( m1_trees_1(C,k1_relat_1(B)) => ( r1_xreal_0(k5_modal_1(k1_relat_1(B),C),2) & ~ ( k5_modal_1(k1_relat_1(B),C) = 0 & k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,0,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,3,D) ) ) & ~ ( k5_modal_1(k1_relat_1(B),C) = 1 & k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,1,0) & k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,1,1) ) & ( k5_modal_1(k1_relat_1(B),C) = 2 => k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) = k1_domain_1(k5_numbers,k5_numbers,2,0) ) ) ) ) ) ) ) ) ), file(modal_1,d6_modal_1), []). fof(d8_modal_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) ) => ! [C] : ( ( v1_funct_1(C) & v3_trees_2(C) & m3_trees_2(C,A) ) => ! [D] : ( m2_finseq_1(D,k5_numbers) => ( r2_hidden(D,k1_relat_1(B)) => k9_modal_1(A,B,C,D) = k8_trees_2(B,D,C) ) ) ) ) ) ), file(modal_1,d8_modal_1), []). fof(d8_xboole_0,axiom,( ! [A,B] : ( r2_xboole_0(A,B) <=> ( r1_tarski(A,B) & A != B ) ) ), file(xboole_0,d8_xboole_0), []). fof(dt_k10_trees_2,axiom,( $true ), file(trees_2,k10_trees_2), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => m1_subset_1(k12_mcart_1(A,B,C,D),k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(mcart_1,k12_mcart_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_enumset1,axiom,( $true ), file(enumset1,k1_enumset1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & m1_subset_1(B,A) ) => m1_subset_1(k1_trees_2(A,B),k1_zfmisc_1(A)) ) ), file(trees_2,k1_trees_2), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). 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_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_trees_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k2_trees_1(A)) & v1_finset_1(k2_trees_1(A)) & v1_trees_1(k2_trees_1(A)) ) ) ), file(trees_1,k2_trees_1), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => m1_subset_1(k3_trees_2(A,B,C),A) ) ), file(trees_2,k3_trees_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_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_modal_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & m1_subset_1(B,A) ) => m2_subset_1(k5_modal_1(A,B),k1_numbers,k5_numbers) ) ), file(modal_1,k5_modal_1), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_trees_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & m1_finseq_1(B,k5_numbers) & ~ v1_xboole_0(C) & v1_trees_1(C) ) => ( ~ v1_xboole_0(k5_trees_1(A,B,C)) & v1_trees_1(k5_trees_1(A,B,C)) ) ) ), file(trees_1,k5_trees_1), []). fof(dt_k6_modal_1,axiom,( m1_modal_1(k6_modal_1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ), file(modal_1,k6_modal_1), []). fof(dt_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k7_domain_1(A,B,C),k1_zfmisc_1(A)) ) ), file(domain_1,k7_domain_1), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k8_trees_2,axiom,( ! [A,B,C] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) & m1_finseq_1(B,k5_numbers) & v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( v1_relat_1(k8_trees_2(A,B,C)) & v1_funct_1(k8_trees_2(A,B,C)) & v3_trees_2(k8_trees_2(A,B,C)) ) ) ), file(trees_2,k8_trees_2), []). fof(dt_k9_finseq_1,axiom,( ! [A] : ( v1_relat_1(k9_finseq_1(A)) & v1_funct_1(k9_finseq_1(A)) ) ), file(finseq_1,k9_finseq_1), []). fof(dt_k9_modal_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & v1_funct_1(C) & v3_trees_2(C) & m3_trees_2(C,A) & m1_finseq_1(D,k5_numbers) ) => ( v1_funct_1(k9_modal_1(A,B,C,D)) & v3_trees_2(k9_modal_1(A,B,C,D)) & m3_trees_2(k9_modal_1(A,B,C,D),A) ) ) ), file(modal_1,k9_modal_1), []). fof(dt_k9_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_trees_1(B) & m1_subset_1(C,A) ) => ( v1_funct_1(k9_trees_2(A,B,C)) & v3_trees_2(k9_trees_2(A,B,C)) & m3_trees_2(k9_trees_2(A,B,C),A) ) ) ), file(trees_2,k9_trees_2), []). 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_modal_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_modal_1(B,A) => ~ v1_xboole_0(B) ) ) ), file(modal_1,m1_modal_1), []). 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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) => m2_finseq_1(B,k5_numbers) ) ) ), file(trees_1,m1_trees_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_modal_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_modal_1(B,A) ) => ! [C] : ( m2_modal_1(C,A,B) => ( v1_funct_1(C) & v3_trees_2(C) & m3_trees_2(C,A) ) ) ) ), file(modal_1,m2_modal_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(dt_m3_trees_2,axiom,( ! [A,B] : ( m3_trees_2(B,A) => v1_relat_1(B) ) ), file(trees_2,m3_trees_2), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_modal_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : m1_modal_1(B,A) ) ), file(modal_1,m1_modal_1), []). 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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : m1_trees_1(B,A) ) ), file(trees_1,m1_trees_1), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_modal_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_modal_1(B,A) ) => ? [C] : m2_modal_1(C,A,B) ) ), file(modal_1,m2_modal_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(existence_m3_trees_2,axiom,( ! [A] : ? [B] : m3_trees_2(B,A) ), file(trees_2,m3_trees_2), []). 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(fc12_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc12_finseq_1), []). fof(fc13_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(A,B)) & v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,fc13_finseq_1), []). fof(fc14_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(B,A)) & v1_relat_1(k7_finseq_1(B,A)) & v1_funct_1(k7_finseq_1(B,A)) & v1_finset_1(k7_finseq_1(B,A)) & v1_finseq_1(k7_finseq_1(B,A)) ) ) ), file(finseq_1,fc14_finseq_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(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(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_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_trees_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & m1_subset_1(B,A) & ~ v1_xboole_0(C) & v1_finset_1(C) & v1_trees_1(C) ) => ( ~ v1_xboole_0(k5_trees_1(A,B,C)) & v1_finset_1(k5_trees_1(A,B,C)) & v1_trees_1(k5_trees_1(A,B,C)) ) ) ), file(trees_1,fc2_trees_1), []). fof(fc2_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ( ~ v1_xboole_0(k1_relat_1(A)) & v1_trees_1(k1_relat_1(A)) ) ) ), file(trees_2,fc2_trees_2), []). fof(fc3_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) ) ), file(finseq_1,fc3_finseq_1), []). fof(fc3_finset_1,axiom,( ! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) ), file(finset_1,fc3_finset_1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) & v3_trees_2(k2_funcop_1(A,B)) ) ) ), file(trees_2,fc3_trees_2), []). fof(fc4_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc4_finseq_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_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc4_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & m1_subset_1(B,A) ) => v1_finset_1(k1_trees_2(A,B)) ) ), file(trees_2,fc4_trees_2), []). fof(fc5_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(trees_2,fc5_trees_2), []). fof(fraenkel_a_1_3_modal_1,axiom,( ! [A,B] : ( m1_trees_1(B,k1_relat_1(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(2),k1_domain_1(k5_numbers,k5_numbers,2,0)))) => ( r2_hidden(A,a_1_3_modal_1(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)) & r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),k1_relat_1(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(2),k1_domain_1(k5_numbers,k5_numbers,2,0)))) ) ) ) ), file(modal_1,a_1_3_modal_1), []). fof(fraenkel_a_2_1_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) & m1_trees_1(C,B) ) => ( r2_hidden(A,a_2_1_trees_2(B,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D)) & r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D)),B) ) ) ) ), file(trees_2,a_2_1_trees_2), []). fof(irreflexivity_r2_xboole_0,axiom,( ! [A,B] : ~ r2_xboole_0(A,A) ), file(xboole_0,r2_xboole_0), []). fof(l1_modal_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r2_xboole_0(k1_xboole_0,k12_finseq_1(k5_numbers,A)) ) ), file(modal_1,l1_modal_1), []). fof(l31_modal_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_finset_1(k1_relat_1(A)) => v1_finset_1(A) ) ) ), file(modal_1,l31_modal_1), []). fof(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). 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_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_trees_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_trees_1(A) ) ), file(trees_1,rc1_trees_1), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). 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_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_trees_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) ) ), file(trees_1,rc2_trees_1), []). fof(rc2_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), []). 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_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), []). fof(rc4_finseq_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_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_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc5_trees_2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) ), file(trees_2,rc5_trees_2), []). 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(rc6_trees_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) ) ) ), file(trees_2,rc6_trees_2), []). 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(rc7_trees_2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v3_trees_2(A) ) ), file(trees_2,rc7_trees_2), []). 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(rc8_trees_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v3_trees_2(B) ) ) ), file(trees_2,rc8_trees_2), []). fof(rc9_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & ~ v1_xboole_0(C) ) ) ), file(trees_2,rc9_trees_2), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => k12_mcart_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ), file(mcart_1,k12_mcart_1), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). fof(redefinition_k3_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => k3_trees_2(A,B,C) = k1_funct_1(B,C) ) ), file(trees_2,k3_trees_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_k5_modal_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & m1_subset_1(B,A) ) => k5_modal_1(A,B) = k10_trees_2(A,B) ) ), file(modal_1,k5_modal_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k2_tarski(B,C) ) ), file(domain_1,k7_domain_1), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_k9_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_trees_1(B) & m1_subset_1(C,A) ) => k9_trees_2(A,B,C) = k2_funcop_1(B,C) ) ), file(trees_2,k9_trees_2), []). fof(redefinition_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) <=> m1_subset_1(B,A) ) ) ), file(trees_1,m1_trees_1), []). 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_modal_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_modal_1(B,A) ) => ! [C] : ( m2_modal_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(modal_1,m2_modal_1), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), []). fof(redefinition_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) <=> r2_tarski(A,B) ) ), file(wellord2,r2_wellord2), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(reflexivity_r2_wellord2,axiom,( ! [A,B] : r2_wellord2(A,A) ), file(wellord2,r2_wellord2), []). fof(reflexivity_r3_xboole_0,axiom,( ! [A,B] : r3_xboole_0(A,A) ), file(xboole_0,r3_xboole_0), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(spc3_boole,axiom,( ~ v1_xboole_0(3) ), file(boole,spc3_boole), []). fof(spc3_numerals,axiom, ( v2_xreal_0(3) & m1_subset_1(3,k5_numbers) ), file(numerals,spc3_numerals), []). fof(symmetry_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) => r2_wellord2(B,A) ) ), file(wellord2,r2_wellord2), []). fof(symmetry_r3_xboole_0,axiom,( ! [A,B] : ( r3_xboole_0(A,B) => r3_xboole_0(B,A) ) ), file(xboole_0,r3_xboole_0), []). fof(t10_modal_1,axiom,( k2_trees_1(2) = k1_enumset1(k1_xboole_0,k12_finseq_1(k5_numbers,0),k12_finseq_1(k5_numbers,1)) ), file(modal_1,t10_modal_1), []). 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(t16_trees_1,axiom,( ! [A,B] : ( r1_tarski(k9_finseq_1(A),k9_finseq_1(B)) <=> A = B ) ), file(trees_1,t16_trees_1), []). fof(t17_modal_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) ) => ! [C] : ( m2_finseq_1(C,k5_numbers) => ( r2_hidden(C,A) => ! [D] : ( m1_trees_1(D,k5_trees_1(A,C,B)) => ! [E] : ( m1_trees_1(E,A) => ( ( D = E & r2_xboole_0(E,C) ) => k1_trees_2(k5_trees_1(A,C,B),D) = k1_trees_2(A,E) ) ) ) ) ) ) ) ), file(modal_1,t17_modal_1), []). fof(t18_modal_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) ) => ! [C] : ( m2_finseq_1(C,k5_numbers) => ( r2_hidden(C,A) => ! [D] : ( m1_trees_1(D,k5_trees_1(A,C,B)) => ! [E] : ( m1_trees_1(E,A) => ( D = E => ( r3_xboole_0(C,E) | k1_trees_2(k5_trees_1(A,C,B),D) = k1_trees_2(A,E) ) ) ) ) ) ) ) ) ), file(modal_1,t18_modal_1), []). fof(t19_funcop_1,axiom,( ! [A,B] : ( k1_relat_1(k2_funcop_1(A,B)) = A & r1_tarski(k2_relat_1(k2_funcop_1(A,B)),k1_tarski(B)) ) ), file(funcop_1,t19_funcop_1), []). fof(t19_modal_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) ) => ! [C] : ( m2_finseq_1(C,k5_numbers) => ( r2_hidden(C,A) => ! [D] : ( m1_trees_1(D,k5_trees_1(A,C,B)) => ! [E] : ( m1_trees_1(E,B) => ( D = k8_finseq_1(k5_numbers,C,E) => r2_tarski(k1_trees_2(k5_trees_1(A,C,B),D),k1_trees_2(B,E)) ) ) ) ) ) ) ) ), file(modal_1,t19_modal_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(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(t3_modal_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_finseq_1(C,k5_numbers) => ~ ( A != B & r3_xboole_0(k12_finseq_1(k5_numbers,A),k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,B),C)) ) ) ) ) ), file(modal_1,t3_modal_1), []). 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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k7_finseq_1(A,k1_xboole_0) = A & k7_finseq_1(k1_xboole_0,A) = A ) ) ), file(finseq_1,t47_finseq_1), []). 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(t55_trees_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(B,A) => r2_hidden(k12_finseq_1(k5_numbers,A),k2_trees_1(B)) ) ) ) ), file(trees_1,t55_trees_1), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t76_card_2,axiom,( ! [A,B] : ( A != B => k4_card_1(k2_tarski(A,B)) = 2 ) ), file(card_2,t76_card_2), []). 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(t81_card_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r2_wellord2(A,B) => k4_card_1(A) = k4_card_1(B) ) ) ) ), file(card_1,t81_card_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(t8_trees_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( r1_tarski(A,B) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) & B = k7_finseq_1(A,C) ) ) ) ) ), file(trees_1,t8_trees_1), []).