% Mizar problem: t30_bciideal,bciideal,769,5 
fof(t30_bciideal, conjecture,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  =>  ( (! [B] :  ( ( ~ (v2_struct_0(B))  &  (v3_bcialg_1(B) &  (v4_bcialg_1(B) &  (v5_bcialg_1(B) &  (v7_bcialg_1(B) & l2_bcialg_1(B)) ) ) ) )  =>  (! [C] :  (m1_subset_1(C, u1_struct_0(B)) =>  (! [D] :  (m1_subset_1(D, u1_struct_0(B)) => k1_bcialg_1(B, k1_bcialg_1(B, C, D), k1_bcialg_1(B, D, C))=k1_bcialg_1(B, C, D)) ) ) ) ) )  => u1_struct_0(A)=k4_bcialg_1(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(cc10_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ( ~ (v2_struct_0(A))  & v7_struct_0(A))  => v13_struct_0(A, 1)) ) ) ).
fof(cc11_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v13_struct_0(A, 1) =>  ( ~ (v2_struct_0(A))  & v7_struct_0(A)) ) ) ) ).
fof(cc17_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ~ (v7_struct_0(A))  =>  ~ (v2_struct_0(A)) ) ) ) ).
fof(cc1_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v2_struct_0(A) => v7_struct_0(A)) ) ) ).
fof(cc2_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ~ (v7_struct_0(A))  =>  ~ (v2_struct_0(A)) ) ) ) ).
fof(cc4_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v2_struct_0(A) =>  (v2_struct_0(A) & v8_struct_0(A)) ) ) ) ).
fof(cc5_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ~ (v8_struct_0(A))  =>  ( ~ (v2_struct_0(A))  &  ~ (v8_struct_0(A)) ) ) ) ) ).
fof(cc6_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v7_struct_0(A) => v8_struct_0(A)) ) ) ).
fof(cc7_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ~ (v8_struct_0(A))  =>  ~ (v7_struct_0(A)) ) ) ) ).
fof(cc8_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v2_struct_0(A) => v13_struct_0(A, k5_ordinal1)) ) ) ).
fof(cc9_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v13_struct_0(A, k5_ordinal1) => v2_struct_0(A)) ) ) ).
fof(d12_bcialg_1, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  => k4_bcialg_1(A)=a_1_0_bcialg_1(A)) ) ).
fof(d1_bcialg_1, axiom,  (! [A] :  (l1_bcialg_1(A) =>  (! [B] :  (m1_subset_1(B, u1_struct_0(A)) =>  (! [C] :  (m1_subset_1(C, u1_struct_0(A)) => k1_bcialg_1(A, B, C)=k4_binop_1(u1_struct_0(A), u1_bcialg_1(A), B, C)) ) ) ) ) ) ).
fof(dt_k1_bcialg_1, axiom,  (! [A, B, C] :  ( (l1_bcialg_1(A) &  (m1_subset_1(B, u1_struct_0(A)) & m1_subset_1(C, u1_struct_0(A))) )  => m1_subset_1(k1_bcialg_1(A, B, C), u1_struct_0(A))) ) ).
fof(dt_k1_binop_1, axiom, $true).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_zfmisc_1, axiom, $true).
fof(dt_k4_bcialg_1, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  =>  ( ~ (v1_xboole_0(k4_bcialg_1(A)))  & m1_subset_1(k4_bcialg_1(A), k1_zfmisc_1(u1_struct_0(A)))) ) ) ).
fof(dt_k4_binop_1, axiom,  (! [A, B, C, D] :  ( ( (v1_funct_1(B) &  (v1_funct_2(B, k2_zfmisc_1(A, A), A) & m1_subset_1(B, k1_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(A, A), A)))) )  &  (m1_subset_1(C, A) & m1_subset_1(D, A)) )  => m1_subset_1(k4_binop_1(A, B, C, D), A)) ) ).
fof(dt_k4_struct_0, axiom,  (! [A] :  (l2_struct_0(A) => m1_subset_1(k4_struct_0(A), u1_struct_0(A))) ) ).
fof(dt_k5_ordinal1, axiom, $true).
fof(dt_l1_bcialg_1, axiom,  (! [A] :  (l1_bcialg_1(A) => l1_struct_0(A)) ) ).
fof(dt_l1_struct_0, axiom, $true).
fof(dt_l2_bcialg_1, axiom,  (! [A] :  (l2_bcialg_1(A) =>  (l1_bcialg_1(A) & l2_struct_0(A)) ) ) ).
fof(dt_l2_struct_0, axiom,  (! [A] :  (l2_struct_0(A) => l1_struct_0(A)) ) ).
fof(dt_m1_subset_1, axiom, $true).
fof(dt_u1_bcialg_1, axiom,  (! [A] :  (l1_bcialg_1(A) =>  (v1_funct_1(u1_bcialg_1(A)) &  (v1_funct_2(u1_bcialg_1(A), k2_zfmisc_1(u1_struct_0(A), u1_struct_0(A)), u1_struct_0(A)) & m1_subset_1(u1_bcialg_1(A), k1_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A), u1_struct_0(A)), u1_struct_0(A))))) ) ) ) ).
fof(dt_u1_struct_0, axiom, $true).
fof(existence_l1_bcialg_1, axiom,  (? [A] : l1_bcialg_1(A)) ).
fof(existence_l1_struct_0, axiom,  (? [A] : l1_struct_0(A)) ).
fof(existence_l2_bcialg_1, axiom,  (? [A] : l2_bcialg_1(A)) ).
fof(existence_l2_struct_0, axiom,  (? [A] : l2_struct_0(A)) ).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(fc10_struct_0, axiom,  (! [A] :  (l2_struct_0(A) => v9_struct_0(k4_struct_0(A), A)) ) ).
fof(fc1_bcialg_2, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  =>  (v10_bcialg_1(k4_struct_0(A), A) & v1_bcialg_2(k4_struct_0(A), A)) ) ) ).
fof(fc1_struct_0, axiom,  (! [A] :  ( (v2_struct_0(A) & l1_struct_0(A))  => v1_xboole_0(u1_struct_0(A))) ) ).
fof(fc2_bcialg_2, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  => v5_bcialg_2(k4_struct_0(A), A)) ) ).
fof(fc2_struct_0, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  & l1_struct_0(A))  =>  ~ (v1_xboole_0(u1_struct_0(A))) ) ) ).
fof(fc6_struct_0, axiom,  (! [A] :  ( ( ~ (v7_struct_0(A))  & l1_struct_0(A))  =>  ~ (v1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(fc7_struct_0, axiom,  (! [A] :  ( (v7_struct_0(A) & l1_struct_0(A))  => v1_zfmisc_1(u1_struct_0(A))) ) ).
fof(fc8_struct_0, axiom,  (! [A] :  ( (v8_struct_0(A) & l1_struct_0(A))  => v1_finset_1(u1_struct_0(A))) ) ).
fof(fc9_struct_0, axiom,  (! [A] :  ( ( ~ (v8_struct_0(A))  & l1_struct_0(A))  =>  ~ (v1_finset_1(u1_struct_0(A))) ) ) ).
fof(fraenkel_a_1_0_bcialg_1, axiom,  (! [A, B] :  ( ( ~ (v2_struct_0(B))  &  (v3_bcialg_1(B) &  (v4_bcialg_1(B) &  (v5_bcialg_1(B) &  (v7_bcialg_1(B) & l2_bcialg_1(B)) ) ) ) )  =>  (r2_hidden(A, a_1_0_bcialg_1(B)) <=>  (? [C] :  (m1_subset_1(C, u1_struct_0(B)) &  (A=C & r1_bcialg_1(B, k4_struct_0(B), C)) ) ) ) ) ) ).
fof(rc17_struct_0, axiom,  (! [A] :  (l2_struct_0(A) =>  (? [B] :  (m1_subset_1(B, u1_struct_0(A)) & v9_struct_0(B, A)) ) ) ) ).
fof(rc19_struct_0, axiom,  (! [A] :  ( ( ~ (v7_struct_0(A))  & l2_struct_0(A))  =>  (? [B] :  (m1_subset_1(B, u1_struct_0(A)) &  ~ (v9_struct_0(B, A)) ) ) ) ) ).
fof(rc1_bcialg_2, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  =>  (? [B] :  (m1_subset_1(B, u1_struct_0(A)) & v1_bcialg_2(B, A)) ) ) ) ).
fof(rc20_struct_0, axiom,  (? [A] :  (l2_struct_0(A) &  ( ~ (v2_struct_0(A))  & v7_struct_0(A)) ) ) ).
fof(rc21_struct_0, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  & l1_struct_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(u1_struct_0(A))) &  ( ~ (v1_xboole_0(B))  & v1_zfmisc_1(B)) ) ) ) ) ).
fof(rc22_struct_0, axiom,  (! [A] :  ( ( ~ (v7_struct_0(A))  & l1_struct_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(u1_struct_0(A))) &  ~ (v1_zfmisc_1(B)) ) ) ) ) ).
fof(rc4_struct_0, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  & l1_struct_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(u1_struct_0(A))) &  ~ (v1_xboole_0(B)) ) ) ) ) ).
fof(redefinition_k4_binop_1, axiom,  (! [A, B, C, D] :  ( ( (v1_funct_1(B) &  (v1_funct_2(B, k2_zfmisc_1(A, A), A) & m1_subset_1(B, k1_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(A, A), A)))) )  &  (m1_subset_1(C, A) & m1_subset_1(D, A)) )  => k4_binop_1(A, B, C, D)=k1_binop_1(B, C, D)) ) ).
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(spc1_boole, axiom,  ~ (v1_xboole_0(1)) ).
fof(t16_bcialg_1, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) )  =>  ( (! [B] :  ( ( ~ (v2_struct_0(B))  &  (v3_bcialg_1(B) &  (v4_bcialg_1(B) &  (v5_bcialg_1(B) &  (v7_bcialg_1(B) & l2_bcialg_1(B)) ) ) ) )  =>  (! [C] :  (m1_subset_1(C, u1_struct_0(B)) =>  (! [D] :  (m1_subset_1(D, u1_struct_0(B)) => k1_bcialg_1(B, k1_bcialg_1(B, C, D), k1_bcialg_1(B, D, C))=k1_bcialg_1(B, C, D)) ) ) ) ) )  =>  ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) &  (v8_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) ) ) ) ) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t25_bciideal, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v3_bcialg_1(A) &  (v4_bcialg_1(A) &  (v5_bcialg_1(A) &  (v7_bcialg_1(A) &  (v8_bcialg_1(A) & l2_bcialg_1(A)) ) ) ) ) )  => k4_bcialg_1(A)=u1_struct_0(A)) ) ).
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(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_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)) ) ) ) ) ).
