% Mizar ND problem: t2_trees_2,trees_2,54,26 fof(dh_c1_2__trees_2,definition, ( ( ( v1_relat_1(c1_2__trees_2) & v1_funct_1(c1_2__trees_2) & v1_finseq_1(c1_2__trees_2) ) => ! [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) ) => ( ( r2_xboole_0(c1_2__trees_2,B) & r1_tarski(A,B) ) => r3_xboole_0(c1_2__trees_2,A) ) ) ) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v1_finseq_1(D) ) => ! [E] : ( ( v1_relat_1(E) & v1_funct_1(E) & v1_finseq_1(E) ) => ( ( r2_xboole_0(C,E) & r1_tarski(D,E) ) => r3_xboole_0(C,D) ) ) ) ) ), introduced(definition,[new_symbol(c1_2__trees_2),file(trees_2,c1_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,c1_2__trees_2)]). fof(dh_c2_2__trees_2,definition, ( ( ( v1_relat_1(c2_2__trees_2) & v1_funct_1(c2_2__trees_2) & v1_finseq_1(c2_2__trees_2) ) => ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( ( r2_xboole_0(c1_2__trees_2,A) & r1_tarski(c2_2__trees_2,A) ) => r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ) ) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( r2_xboole_0(c1_2__trees_2,C) & r1_tarski(B,C) ) => r3_xboole_0(c1_2__trees_2,B) ) ) ) ), introduced(definition,[new_symbol(c2_2__trees_2),file(trees_2,c2_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,c2_2__trees_2)]). fof(dh_c3_2__trees_2,definition, ( ( ( v1_relat_1(c3_2__trees_2) & v1_funct_1(c3_2__trees_2) & v1_finseq_1(c3_2__trees_2) ) => ( ( r2_xboole_0(c1_2__trees_2,c3_2__trees_2) & r1_tarski(c2_2__trees_2,c3_2__trees_2) ) => r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ) ) => ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( ( r2_xboole_0(c1_2__trees_2,A) & r1_tarski(c2_2__trees_2,A) ) => r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ) ) ), introduced(definition,[new_symbol(c3_2__trees_2),file(trees_2,c3_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,c3_2__trees_2)]). fof(e1_2__trees_2,assumption,( r2_xboole_0(c1_2__trees_2,c3_2__trees_2) ), introduced(assumption,[file(trees_2,e1_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,e1_2__trees_2)]). fof(rc3_finseq_1,theorem,( ? [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), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_relat_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) ) ), file(relat_1,rc3_relat_1), [interesting(0.9),axiom,file(relat_1,rc3_relat_1)]). fof(rc6_finseq_1,theorem,( ? [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), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(fc12_relat_1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) ), file(relat_1,fc12_relat_1), [interesting(0.9),axiom,file(relat_1,fc12_relat_1)]). fof(fc1_xboole_0,theorem,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]). fof(fc2_finseq_1,theorem, ( 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), [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]). fof(fc4_relat_1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) ), file(relat_1,fc4_relat_1), [interesting(0.9),axiom,file(relat_1,fc4_relat_1)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(cc1_finset_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]). fof(cc1_relat_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_relat_1(A) ) ), file(relat_1,cc1_relat_1), [interesting(0.9),axiom,file(relat_1,cc1_relat_1)]). fof(rc1_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]). fof(rc1_relat_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc1_relat_1), [interesting(0.9),axiom,file(relat_1,rc1_relat_1)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc1_xboole_0,theorem,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]). fof(rc2_relat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc2_relat_1), [interesting(0.9),axiom,file(relat_1,rc2_relat_1)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(rc2_xboole_0,theorem,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]). fof(rc3_finset_1,theorem,( ! [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), [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]). fof(rc4_finset_1,theorem,( ! [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), [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(rc8_finseq_1,theorem,( ? [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), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(cc2_finset_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]). fof(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(symmetry_r3_xboole_0,theorem,( ! [A,B] : ( r3_xboole_0(A,B) => r3_xboole_0(B,A) ) ), file(xboole_0,r3_xboole_0), [interesting(0.9),axiom,file(xboole_0,r3_xboole_0)]). fof(reflexivity_r3_xboole_0,theorem,( ! [A,B] : r3_xboole_0(A,A) ), file(xboole_0,r3_xboole_0), [interesting(0.9),axiom,file(xboole_0,r3_xboole_0)]). fof(dt_c1_2__trees_2,assumption, ( v1_relat_1(c1_2__trees_2) & v1_funct_1(c1_2__trees_2) & v1_finseq_1(c1_2__trees_2) ), introduced(assumption,[file(trees_2,c1_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,c1_2__trees_2)]). fof(dt_c2_2__trees_2,assumption, ( v1_relat_1(c2_2__trees_2) & v1_funct_1(c2_2__trees_2) & v1_finseq_1(c2_2__trees_2) ), introduced(assumption,[file(trees_2,c2_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,c2_2__trees_2)]). fof(dt_c3_2__trees_2,assumption, ( v1_relat_1(c3_2__trees_2) & v1_funct_1(c3_2__trees_2) & v1_finseq_1(c3_2__trees_2) ), introduced(assumption,[file(trees_2,c3_2__trees_2)]), [interesting(0.8),axiom,file(trees_2,c3_2__trees_2)]). fof(cc1_finseq_1,theorem,( ! [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), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(irreflexivity_r2_xboole_0,theorem,( ! [A,B] : ~ r2_xboole_0(A,A) ), file(xboole_0,r2_xboole_0), [interesting(0.9),axiom,file(xboole_0,r2_xboole_0)]). fof(antisymmetry_r2_xboole_0,theorem,( ! [A,B] : ( r2_xboole_0(A,B) => ~ r2_xboole_0(B,A) ) ), file(xboole_0,r2_xboole_0), [interesting(0.9),axiom,file(xboole_0,r2_xboole_0)]). fof(d8_xboole_0,definition,( ! [A,B] : ( r2_xboole_0(A,B) <=> ( r1_tarski(A,B) & A != B ) ) ), file(xboole_0,d8_xboole_0), [interesting(0.9),axiom,file(xboole_0,d8_xboole_0)]). fof(e2_2__trees_2,plain,( r1_tarski(c1_2__trees_2,c3_2__trees_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__trees_2,dt_c3_2__trees_2,e1_2__trees_2])],[rc3_finseq_1,rc3_relat_1,rc6_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc2_finseq_1,fc4_relat_1,t1_subset,t4_subset,t5_subset,cc1_finset_1,cc1_relat_1,cc2_finset_1,rc1_finset_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_relat_1,rc2_subset_1,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_subset_1,cc1_finseq_1,fc1_subset_1,rc1_finseq_1,reflexivity_r1_tarski,irreflexivity_r2_xboole_0,antisymmetry_r2_xboole_0,dt_c1_2__trees_2,dt_c3_2__trees_2,t3_subset,e1_2__trees_2,d8_xboole_0]), [interesting(0.8),file(trees_2,e2_2__trees_2),[file(trees_2,e2_2__trees_2)]]). fof(t1_trees_2,theorem,( ! [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) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( r1_tarski(A,C) & r1_tarski(B,C) ) => r3_xboole_0(A,B) ) ) ) ) ), file(trees_2,t1_trees_2), [interesting(0.9),axiom,file(trees_2,t1_trees_2)]). fof(e3_2__trees_2,plain,( ~ ( r1_tarski(c2_2__trees_2,c3_2__trees_2) & ~ r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ) ), inference(mizar_by,[status(thm),assumptions([dt_c2_2__trees_2,dt_c1_2__trees_2,dt_c3_2__trees_2,e1_2__trees_2])],[rc3_finseq_1,rc3_relat_1,rc6_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc2_finseq_1,fc4_relat_1,t1_subset,t4_subset,t5_subset,t8_boole,cc1_finset_1,cc1_relat_1,rc1_finset_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_relat_1,rc2_subset_1,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t6_boole,t7_boole,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_subset_1,cc2_finset_1,fc1_subset_1,reflexivity_r1_tarski,symmetry_r3_xboole_0,reflexivity_r3_xboole_0,dt_c1_2__trees_2,dt_c2_2__trees_2,dt_c3_2__trees_2,cc1_finseq_1,rc1_finseq_1,t3_subset,e2_2__trees_2,t1_trees_2]), [interesting(0.8),file(trees_2,e3_2__trees_2),[file(trees_2,e3_2__trees_2)]]). fof(i3_2__trees_2,theorem,( $true ), introduced(tautology,[file(trees_2,i3_2__trees_2)]), [interesting(0.8),trivial,file(trees_2,i3_2__trees_2)]). fof(i2_2__trees_2,plain,( ~ ( r1_tarski(c2_2__trees_2,c3_2__trees_2) & ~ r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ) ), inference(conclusion,[status(thm),assumptions([dt_c2_2__trees_2,dt_c1_2__trees_2,dt_c3_2__trees_2,e1_2__trees_2])],[e3_2__trees_2,i3_2__trees_2]), [interesting(0.8),file(trees_2,i2_2__trees_2),[file(trees_2,i2_2__trees_2)]]). fof(i1_2__trees_2,plain, ( ( r2_xboole_0(c1_2__trees_2,c3_2__trees_2) & r1_tarski(c2_2__trees_2,c3_2__trees_2) ) => r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_2__trees_2,dt_c1_2__trees_2,dt_c3_2__trees_2]),discharge_asm(discharge,[e1_2__trees_2])],[e1_2__trees_2,i2_2__trees_2]), [interesting(0.8),file(trees_2,i1_2__trees_2),[file(trees_2,i1_2__trees_2)]]). fof(i1_2_tmp__trees_2,plain, ( ( v1_relat_1(c1_2__trees_2) & v1_funct_1(c1_2__trees_2) & v1_finseq_1(c1_2__trees_2) & v1_relat_1(c2_2__trees_2) & v1_funct_1(c2_2__trees_2) & v1_finseq_1(c2_2__trees_2) & v1_relat_1(c3_2__trees_2) & v1_funct_1(c3_2__trees_2) & v1_finseq_1(c3_2__trees_2) ) => ( ( r2_xboole_0(c1_2__trees_2,c3_2__trees_2) & r1_tarski(c2_2__trees_2,c3_2__trees_2) ) => r3_xboole_0(c1_2__trees_2,c2_2__trees_2) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__trees_2,dt_c2_2__trees_2,dt_c3_2__trees_2])],[dt_c1_2__trees_2,dt_c2_2__trees_2,dt_c3_2__trees_2,i1_2__trees_2]), [interesting(1),t2_trees_2]). fof(t2_trees_2,theorem,( ! [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) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( r2_xboole_0(A,C) & r1_tarski(B,C) ) => r3_xboole_0(A,B) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_2_tmp__trees_2,dh_c1_2__trees_2,dh_c2_2__trees_2,dh_c3_2__trees_2]), [interesting(1),file(trees_2,t2_trees_2),[file(trees_2,t2_trees_2)]]).