% Mizar ND problem: t3_grcat_1,grcat_1,52,55 fof(dh_c1_2__grcat_1,definition, ( ( ( ~ v1_xboole_0(c1_2__grcat_1) & v1_classes2(c1_2__grcat_1) ) => ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ! [B] : ( m1_subset_1(B,c1_2__grcat_1) => ! [C] : ( m1_subset_1(C,c1_2__grcat_1) => ! [D] : ( m1_subset_1(D,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(A,B,C),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(A,B,C,D),c1_2__grcat_1) ) ) ) ) ) ) => ! [E] : ( ( ~ v1_xboole_0(E) & v1_classes2(E) ) => ! [F] : ( m1_subset_1(F,E) => ! [G] : ( m1_subset_1(G,E) => ! [H] : ( m1_subset_1(H,E) => ! [I] : ( m1_subset_1(I,E) => ( m1_subset_1(k3_mcart_1(F,G,H),E) & m1_subset_1(k4_mcart_1(F,G,H,I),E) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_2__grcat_1),file(grcat_1,c1_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c1_2__grcat_1)]). fof(dh_c2_2__grcat_1,definition, ( ( m1_subset_1(c2_2__grcat_1,c1_2__grcat_1) => ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ! [B] : ( m1_subset_1(B,c1_2__grcat_1) => ! [C] : ( m1_subset_1(C,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,A,B),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,A,B,C),c1_2__grcat_1) ) ) ) ) ) => ! [D] : ( m1_subset_1(D,c1_2__grcat_1) => ! [E] : ( m1_subset_1(E,c1_2__grcat_1) => ! [F] : ( m1_subset_1(F,c1_2__grcat_1) => ! [G] : ( m1_subset_1(G,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(D,E,F),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(D,E,F,G),c1_2__grcat_1) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_2__grcat_1),file(grcat_1,c2_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c2_2__grcat_1)]). fof(dh_c3_2__grcat_1,definition, ( ( m1_subset_1(c3_2__grcat_1,c1_2__grcat_1) => ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ! [B] : ( m1_subset_1(B,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,A),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,A,B),c1_2__grcat_1) ) ) ) ) => ! [C] : ( m1_subset_1(C,c1_2__grcat_1) => ! [D] : ( m1_subset_1(D,c1_2__grcat_1) => ! [E] : ( m1_subset_1(E,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,C,D),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,C,D,E),c1_2__grcat_1) ) ) ) ) ), introduced(definition,[new_symbol(c3_2__grcat_1),file(grcat_1,c3_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c3_2__grcat_1)]). fof(dh_c4_2__grcat_1,definition, ( ( m1_subset_1(c4_2__grcat_1,c1_2__grcat_1) => ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,A),c1_2__grcat_1) ) ) ) => ! [B] : ( m1_subset_1(B,c1_2__grcat_1) => ! [C] : ( m1_subset_1(C,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,B),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,B,C),c1_2__grcat_1) ) ) ) ), introduced(definition,[new_symbol(c4_2__grcat_1),file(grcat_1,c4_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c4_2__grcat_1)]). fof(dh_c5_2__grcat_1,definition, ( ( m1_subset_1(c5_2__grcat_1,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,c5_2__grcat_1),c1_2__grcat_1) ) ) => ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,A),c1_2__grcat_1) ) ) ), introduced(definition,[new_symbol(c5_2__grcat_1),file(grcat_1,c5_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c5_2__grcat_1)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(cc1_classes2,theorem,( ! [A] : ( v2_classes1(A) => v1_classes1(A) ) ), file(classes2,cc1_classes2), [interesting(0.9),axiom,file(classes2,cc1_classes2)]). fof(cc3_classes2,theorem,( ! [A] : ( ( v1_ordinal1(A) & v2_classes1(A) ) => v1_classes2(A) ) ), file(classes2,cc3_classes2), [interesting(0.9),axiom,file(classes2,cc3_classes2)]). 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(rc1_classes2,theorem,( ? [A] : ( v1_ordinal1(A) & ~ v1_xboole_0(A) & v1_classes1(A) & v2_classes1(A) & v1_classes2(A) ) ), file(classes2,rc1_classes2), [interesting(0.9),axiom,file(classes2,rc1_classes2)]). fof(commutativity_k2_tarski,theorem,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). 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(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(redefinition_k6_classes2,definition,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k6_classes2(A,B,C) = k4_tarski(B,C) ) ), file(classes2,k6_classes2), [interesting(0.9),axiom,file(classes2,k6_classes2)]). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(dt_k3_mcart_1,axiom,( $true ), file(mcart_1,k3_mcart_1), [interesting(0.9),axiom,file(mcart_1,k3_mcart_1)]). fof(dt_k4_mcart_1,axiom,( $true ), file(mcart_1,k4_mcart_1), [interesting(0.9),axiom,file(mcart_1,k4_mcart_1)]). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), [interesting(0.9),axiom,file(tarski,k4_tarski)]). fof(dt_k6_classes2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k6_classes2(A,B,C),A) ) ), file(classes2,k6_classes2), [interesting(0.9),axiom,file(classes2,k6_classes2)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_c1_2__grcat_1,assumption, ( ~ v1_xboole_0(c1_2__grcat_1) & v1_classes2(c1_2__grcat_1) ), introduced(assumption,[file(grcat_1,c1_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c1_2__grcat_1)]). fof(dt_c2_2__grcat_1,assumption,( m1_subset_1(c2_2__grcat_1,c1_2__grcat_1) ), introduced(assumption,[file(grcat_1,c2_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c2_2__grcat_1)]). fof(dt_c3_2__grcat_1,assumption,( m1_subset_1(c3_2__grcat_1,c1_2__grcat_1) ), introduced(assumption,[file(grcat_1,c3_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c3_2__grcat_1)]). fof(dt_c4_2__grcat_1,assumption,( m1_subset_1(c4_2__grcat_1,c1_2__grcat_1) ), introduced(assumption,[file(grcat_1,c4_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c4_2__grcat_1)]). fof(dt_c5_2__grcat_1,assumption,( m1_subset_1(c5_2__grcat_1,c1_2__grcat_1) ), introduced(assumption,[file(grcat_1,c5_2__grcat_1)]), [interesting(0.8),axiom,file(grcat_1,c5_2__grcat_1)]). fof(cc2_classes2,theorem,( ! [A] : ( v1_classes2(A) => ( v1_ordinal1(A) & v1_classes1(A) & v2_classes1(A) ) ) ), file(classes2,cc2_classes2), [interesting(0.9),axiom,file(classes2,cc2_classes2)]). fof(fc3_subset_1,theorem,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), [interesting(0.9),axiom,file(subset_1,fc3_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_classes2,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ? [B] : ( m1_subset_1(B,A) & ~ v1_xboole_0(B) ) ) ), file(classes2,rc2_classes2), [interesting(0.9),axiom,file(classes2,rc2_classes2)]). 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(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(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(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(d4_mcart_1,definition,( ! [A,B,C,D] : k4_mcart_1(A,B,C,D) = k4_tarski(k3_mcart_1(A,B,C),D) ), file(mcart_1,d4_mcart_1), [interesting(0.9),axiom,file(mcart_1,d4_mcart_1)]). fof(e2_2__grcat_1,plain,( k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,c5_2__grcat_1) = k4_tarski(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c5_2__grcat_1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__grcat_1,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1,dt_c5_2__grcat_1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_classes2,cc3_classes2,fc1_xboole_0,rc1_classes2,t1_subset,cc2_classes2,rc1_xboole_0,rc2_classes2,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,dt_m1_subset_1,dt_c1_2__grcat_1,dt_k3_mcart_1,dt_k4_mcart_1,dt_k4_tarski,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1,dt_c5_2__grcat_1,d4_mcart_1]), [interesting(0.8),file(grcat_1,e2_2__grcat_1),[file(grcat_1,e2_2__grcat_1)]]). fof(d3_mcart_1,definition,( ! [A,B,C] : k3_mcart_1(A,B,C) = k4_tarski(k4_tarski(A,B),C) ), file(mcart_1,d3_mcart_1), [interesting(0.9),axiom,file(mcart_1,d3_mcart_1)]). fof(e1_2__grcat_1,plain,( k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1) = k6_classes2(c1_2__grcat_1,k6_classes2(c1_2__grcat_1,c2_2__grcat_1,c3_2__grcat_1),c4_2__grcat_1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__grcat_1,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_classes2,cc3_classes2,fc1_xboole_0,rc1_classes2,t1_subset,existence_m1_subset_1,dt_m1_subset_1,cc2_classes2,rc1_xboole_0,rc2_classes2,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,redefinition_k6_classes2,dt_k3_mcart_1,dt_k4_tarski,dt_k6_classes2,dt_c1_2__grcat_1,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1,d3_mcart_1]), [interesting(0.8),file(grcat_1,e1_2__grcat_1),[file(grcat_1,e1_2__grcat_1)]]). fof(t64_classes2,theorem,( ! [A,B,C] : ( ( ~ v1_xboole_0(C) & v1_classes2(C) ) => ( ( r2_hidden(A,C) & r2_hidden(B,C) ) => ( r2_hidden(k2_tarski(A,B),C) & r2_hidden(k4_tarski(A,B),C) ) ) ) ), file(classes2,t64_classes2), [interesting(0.9),axiom,file(classes2,t64_classes2)]). fof(e3_2__grcat_1,plain, ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,c5_2__grcat_1),c1_2__grcat_1) ), inference(mizar_by,[status(thm),assumptions([dt_c5_2__grcat_1,dt_c1_2__grcat_1,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1])],[dt_k1_xboole_0,cc1_classes2,cc3_classes2,fc1_xboole_0,rc1_classes2,commutativity_k2_tarski,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k6_classes2,dt_k2_tarski,dt_k3_mcart_1,dt_k4_mcart_1,dt_k4_tarski,dt_k6_classes2,dt_m1_subset_1,dt_c1_2__grcat_1,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1,dt_c5_2__grcat_1,cc2_classes2,fc3_subset_1,rc1_xboole_0,rc2_classes2,rc2_xboole_0,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,e2_2__grcat_1,e1_2__grcat_1,t64_classes2]), [interesting(0.8),file(grcat_1,e3_2__grcat_1),[file(grcat_1,e3_2__grcat_1)]]). fof(i3_2__grcat_1,theorem,( $true ), introduced(tautology,[file(grcat_1,i3_2__grcat_1)]), [interesting(0.8),trivial,file(grcat_1,i3_2__grcat_1)]). fof(i2_2__grcat_1,plain, ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,c5_2__grcat_1),c1_2__grcat_1) ), inference(conclusion,[status(thm),assumptions([dt_c5_2__grcat_1,dt_c1_2__grcat_1,dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1])],[e3_2__grcat_1,i3_2__grcat_1]), [interesting(0.8),file(grcat_1,i2_2__grcat_1),[file(grcat_1,i2_2__grcat_1)]]). fof(i2_2_tmp__grcat_1,plain, ( ( m1_subset_1(c2_2__grcat_1,c1_2__grcat_1) & m1_subset_1(c3_2__grcat_1,c1_2__grcat_1) & m1_subset_1(c4_2__grcat_1,c1_2__grcat_1) & m1_subset_1(c5_2__grcat_1,c1_2__grcat_1) ) => ( m1_subset_1(k3_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(c2_2__grcat_1,c3_2__grcat_1,c4_2__grcat_1,c5_2__grcat_1),c1_2__grcat_1) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__grcat_1]),discharge_asm(discharge,[dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1,dt_c5_2__grcat_1])],[dt_c2_2__grcat_1,dt_c3_2__grcat_1,dt_c4_2__grcat_1,dt_c5_2__grcat_1,i2_2__grcat_1]), [interesting(0.8),i1_2__grcat_1]). fof(i1_2__grcat_1,plain,( ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ! [B] : ( m1_subset_1(B,c1_2__grcat_1) => ! [C] : ( m1_subset_1(C,c1_2__grcat_1) => ! [D] : ( m1_subset_1(D,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(A,B,C),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(A,B,C,D),c1_2__grcat_1) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_2__grcat_1])],[i2_2_tmp__grcat_1,dh_c2_2__grcat_1,dh_c3_2__grcat_1,dh_c4_2__grcat_1,dh_c5_2__grcat_1]), [interesting(0.8),file(grcat_1,i1_2__grcat_1),[file(grcat_1,i1_2__grcat_1)]]). fof(i1_2_tmp__grcat_1,plain, ( ( ~ v1_xboole_0(c1_2__grcat_1) & v1_classes2(c1_2__grcat_1) ) => ! [A] : ( m1_subset_1(A,c1_2__grcat_1) => ! [B] : ( m1_subset_1(B,c1_2__grcat_1) => ! [C] : ( m1_subset_1(C,c1_2__grcat_1) => ! [D] : ( m1_subset_1(D,c1_2__grcat_1) => ( m1_subset_1(k3_mcart_1(A,B,C),c1_2__grcat_1) & m1_subset_1(k4_mcart_1(A,B,C,D),c1_2__grcat_1) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__grcat_1])],[dt_c1_2__grcat_1,i1_2__grcat_1]), [interesting(1),t3_grcat_1]). fof(t3_grcat_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_classes2(A) ) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => ! [E] : ( m1_subset_1(E,A) => ( m1_subset_1(k3_mcart_1(B,C,D),A) & m1_subset_1(k4_mcart_1(B,C,D,E),A) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_2_tmp__grcat_1,dh_c1_2__grcat_1]), [interesting(1),file(grcat_1,t3_grcat_1),[file(grcat_1,t3_grcat_1)]]).