% Mizar problem: t20_mod_2,mod_2,586,5 
fof(t20_mod_2, conjecture,  (! [A] :  ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  =>  (! [B] :  (m1_subset_1(B, A) => m1_subset_1(k1_ordinal1(B), A)) ) ) ) ).
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_classes2, axiom,  (! [A] :  (v2_classes1(A) => v1_classes1(A)) ) ).
fof(cc2_classes2, axiom,  (! [A] :  (v1_classes2(A) =>  (v1_ordinal1(A) & v2_classes1(A)) ) ) ).
fof(cc3_classes2, axiom,  (! [A] :  ( (v1_ordinal1(A) & v2_classes1(A))  => v1_classes2(A)) ) ).
fof(commutativity_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, B)=k2_xboole_0(B, A)) ).
fof(commutativity_k7_classes2, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k7_classes2(A, B, C)=k7_classes2(A, C, B)) ) ).
fof(d1_ordinal1, axiom,  (! [A] : k1_ordinal1(A)=k2_xboole_0(A, k1_tarski(A))) ).
fof(dt_k1_classes2, axiom,  (! [A, B] :  ( ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  & m1_subset_1(B, A))  => m1_subset_1(k1_classes2(A, B), A)) ) ).
fof(dt_k1_ordinal1, axiom, $true).
fof(dt_k1_tarski, axiom, $true).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k2_xboole_0, axiom, $true).
fof(dt_k7_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(k7_classes2(A, B, C), A)) ) ).
fof(dt_m1_subset_1, axiom, $true).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(fc1_xboole_0, axiom, v1_xboole_0(k1_xboole_0)).
fof(fc2_xboole_0, axiom,  (! [A] :  ~ (v1_xboole_0(k1_tarski(A))) ) ).
fof(fc4_xboole_0, axiom,  (! [A, B] :  ( ~ (v1_xboole_0(A))  =>  ~ (v1_xboole_0(k2_xboole_0(A, B))) ) ) ).
fof(fc5_xboole_0, axiom,  (! [A, B] :  ( ~ (v1_xboole_0(A))  =>  ~ (v1_xboole_0(k2_xboole_0(B, A))) ) ) ).
fof(idempotence_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, A)=A) ).
fof(idempotence_k7_classes2, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k7_classes2(A, B, B)=B) ) ).
fof(rc1_classes2, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v1_classes2(A)) ) ).
fof(rc1_xboole_0, axiom,  (? [A] : v1_xboole_0(A)) ).
fof(rc2_classes2, axiom,  (! [A] :  ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  =>  (? [B] :  (m1_subset_1(B, A) &  ~ (v1_xboole_0(B)) ) ) ) ) ).
fof(rc2_xboole_0, axiom,  (? [A] :  ~ (v1_xboole_0(A)) ) ).
fof(redefinition_k1_classes2, axiom,  (! [A, B] :  ( ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  & m1_subset_1(B, A))  => k1_classes2(A, B)=k1_tarski(B)) ) ).
fof(redefinition_k7_classes2, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v1_classes2(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k7_classes2(A, B, C)=k2_xboole_0(B, C)) ) ).
fof(redefinition_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) <=> r2_hidden(A, B)) ) ).
fof(t1_boole, axiom,  (! [A] : k2_xboole_0(A, k1_xboole_0)=A) ).
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(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)) ) ) ) ) ).
