% Mizar problem: t34_domain_1,domain_1,640,5 
fof(t34_domain_1, conjecture,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => k5_subset_1(A, B, C)=a_3_7_domain_1(A, B, C)) ) ) ) ) ) ).
fof(antisymmetry_r2_hidden, axiom,  (! [A, B] :  (r2_hidden(A, B) =>  ~ (r2_hidden(B, A)) ) ) ).
fof(asymmetry_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) =>  ~ (r2_tarski(B, A)) ) ) ).
fof(cc1_relat_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_relat_1(A)) ) ).
fof(cc1_subset_1, axiom,  (! [A] :  (v1_xboole_0(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_xboole_0(B)) ) ) ) ).
fof(cc2_relat_1, axiom,  (! [A] :  (v1_relat_1(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_relat_1(B)) ) ) ) ).
fof(cc2_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  ( ~ (v1_subset_1(B, A))  =>  ~ (v1_xboole_0(B)) ) ) ) ) ) ).
fof(cc3_relat_1, axiom,  (! [A] :  ( (v1_xboole_0(A) & v1_relat_1(A))  =>  (v1_relat_1(A) & v3_relat_1(A)) ) ) ).
fof(cc3_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (v1_xboole_0(B) => v1_subset_1(B, A)) ) ) ) ) ).
fof(cc4_relat_1, axiom,  (! [A] :  ( (v1_xboole_0(A) & v1_relat_1(A))  =>  (v1_relat_1(A) & v2_relat_1(A)) ) ) ).
fof(cc4_subset_1, axiom,  (! [A] :  (v1_xboole_0(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  ~ (v1_subset_1(B, A)) ) ) ) ) ).
fof(commutativity_k5_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => k5_subset_1(A, B, C)=k5_subset_1(A, C, B)) ) ).
fof(commutativity_k5_xboole_0, axiom,  (! [A, B] : k5_xboole_0(A, B)=k5_xboole_0(B, A)) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k5_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => m1_subset_1(k5_subset_1(A, B, C), k1_zfmisc_1(A))) ) ).
fof(dt_k5_xboole_0, axiom, $true).
fof(dt_m1_subset_1, axiom, $true).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(fc1_subset_1, axiom,  (! [A] :  ~ (v1_xboole_0(k1_zfmisc_1(A))) ) ).
fof(fc1_xboole_0, axiom, v1_xboole_0(k1_xboole_0)).
fof(fc4_relat_1, axiom,  (! [A, B] :  ( (v1_relat_1(A) & v1_relat_1(B))  => v1_relat_1(k5_xboole_0(A, B))) ) ).
fof(fraenkel_a_3_5_domain_1, axiom,  (! [A, B, C, D] :  ( ( ~ (v1_xboole_0(B))  &  (m1_subset_1(C, k1_zfmisc_1(B)) & m1_subset_1(D, k1_zfmisc_1(B))) )  =>  (r2_hidden(A, a_3_5_domain_1(B, C, D)) <=>  (? [E] :  (m1_subset_1(E, B) &  (A=E &  ( ~ (r2_tarski(E, C))  <=> r2_tarski(E, D)) ) ) ) ) ) ) ).
fof(fraenkel_a_3_7_domain_1, axiom,  (! [A, B, C, D] :  ( ( ~ (v1_xboole_0(B))  &  (m1_subset_1(C, k1_zfmisc_1(B)) & m1_subset_1(D, k1_zfmisc_1(B))) )  =>  (r2_hidden(A, a_3_7_domain_1(B, C, D)) <=>  (? [E] :  (m1_subset_1(E, B) &  (A=E &  ~ ( (r2_tarski(E, C) <=> r2_tarski(E, D)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_4_4_domain_1, axiom,  (! [A, B, C, D, E] :  ( ( ~ (v1_xboole_0(B))  &  (m1_subset_1(C, k1_zfmisc_1(B)) &  (m1_subset_1(D, k1_zfmisc_1(B)) &  ~ (v1_xboole_0(E)) ) ) )  =>  (r2_hidden(A, a_4_4_domain_1(B, C, D, E)) <=>  (? [F] :  (m1_subset_1(F, E) &  (A=F &  ( ~ (r2_tarski(F, C))  <=> r2_tarski(F, D)) ) ) ) ) ) ) ).
fof(fraenkel_a_4_6_domain_1, axiom,  (! [A, B, C, D, E] :  ( ( ~ (v1_xboole_0(B))  &  (m1_subset_1(C, k1_zfmisc_1(B)) &  (m1_subset_1(D, k1_zfmisc_1(B)) &  ~ (v1_xboole_0(E)) ) ) )  =>  (r2_hidden(A, a_4_6_domain_1(B, C, D, E)) <=>  (? [F] :  (m1_subset_1(F, E) &  (A=F &  ~ ( (r2_tarski(F, C) <=> r2_tarski(F, D)) ) ) ) ) ) ) ) ).
fof(rc1_relat_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v1_relat_1(A)) ) ).
fof(rc1_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) &  ~ (v1_xboole_0(B)) ) ) ) ) ).
fof(rc1_xboole_0, axiom,  (? [A] : v1_xboole_0(A)) ).
fof(rc2_relat_1, axiom,  (? [A] :  (v1_relat_1(A) & v2_relat_1(A)) ) ).
fof(rc2_subset_1, axiom,  (! [A] :  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) & v1_xboole_0(B)) ) ) ).
fof(rc2_xboole_0, axiom,  (? [A] :  ~ (v1_xboole_0(A)) ) ).
fof(rc3_subset_1, axiom,  (! [A] :  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) &  ~ (v1_subset_1(B, A)) ) ) ) ).
fof(rc4_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) & v1_subset_1(B, A)) ) ) ) ).
fof(redefinition_k5_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => k5_subset_1(A, B, C)=k5_xboole_0(B, C)) ) ).
fof(redefinition_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) <=> r2_hidden(A, B)) ) ).
fof(reflexivity_r1_tarski, axiom,  (! [A, B] : r1_tarski(A, A)) ).
fof(s6_domain_1__e3_36__domain_1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A))) )  =>  (! [D] :  ( ~ (v1_xboole_0(D))  =>  ( (! [E] :  (m1_subset_1(E, D) =>  ( ( ~ (r2_tarski(E, B))  <=> r2_tarski(E, C))  <=>  ~ ( (r2_tarski(E, B) <=> r2_tarski(E, C)) ) ) ) )  => a_4_4_domain_1(A, B, C, D)=a_4_6_domain_1(A, B, C, D)) ) ) ) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t2_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, B) =>  (v1_xboole_0(B) | r2_tarski(A, B)) ) ) ) ).
fof(t2_tarski, axiom,  (! [A] :  (! [B] :  ( (! [C] :  (r2_hidden(C, A) <=> r2_hidden(C, B)) )  => A=B) ) ) ).
fof(t32_domain_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => k5_subset_1(A, B, C)=a_3_5_domain_1(A, B, C)) ) ) ) ) ) ).
fof(t3_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, k1_zfmisc_1(B)) <=> r1_tarski(A, B)) ) ) ).
fof(t4_subset, axiom,  (! [A] :  (! [B] :  (! [C] :  ( (r2_tarski(A, B) & m1_subset_1(B, k1_zfmisc_1(C)))  => m1_subset_1(A, C)) ) ) ) ).
fof(t5_boole, axiom,  (! [A] : k5_xboole_0(A, k1_xboole_0)=A) ).
fof(t5_subset, axiom,  (! [A] :  (! [B] :  (! [C] :  ~ ( (r2_tarski(A, B) &  (m1_subset_1(B, k1_zfmisc_1(C)) & v1_xboole_0(C)) ) ) ) ) ) ).
fof(t6_boole, axiom,  (! [A] :  (v1_xboole_0(A) => A=k1_xboole_0) ) ).
fof(t7_boole, axiom,  (! [A] :  (! [B] :  ~ ( (r2_tarski(A, B) & v1_xboole_0(B)) ) ) ) ).
fof(t8_boole, axiom,  (! [A] :  (! [B] :  ~ ( (v1_xboole_0(A) &  ( ~ (A=B)  & v1_xboole_0(B)) ) ) ) ) ).
