% Mizar problem: t2_incproj,incproj,154,15 
fof(t2_incproj, conjecture,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  =>  (u1_incsp_1(k3_incproj(A))=u1_struct_0(A) &  (u2_incsp_1(k3_incproj(A))=k1_incproj(A) & u3_incsp_1(k3_incproj(A))=k2_incproj(A)) ) ) ) ).
fof(existence_m2_collsp, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  =>  (? [B] : m2_collsp(B, A)) ) ) ).
fof(dt_m2_collsp, axiom, $true).
fof(cc11_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v13_struct_0(A, 1) =>  ( ~ (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(spc1_boole, axiom,  ~ (v1_xboole_0(1)) ).
fof(reflexivity_r1_tarski, axiom,  (! [A, B] : r1_tarski(A, 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(existence_m1_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  =>  (? [B] : m1_incproj(B, A)) ) ) ).
fof(redefinition_m1_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  =>  (! [B] :  (m1_incproj(B, A) <=> m2_collsp(B, A)) ) ) ) ).
fof(redefinition_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) <=> r2_hidden(A, B)) ) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k5_ordinal1, axiom, $true).
fof(dt_m1_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  =>  (! [B] :  (m1_incproj(B, A) => m1_subset_1(B, k1_zfmisc_1(u1_struct_0(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(cc17_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ~ (v7_struct_0(A))  =>  ~ (v2_struct_0(A)) ) ) ) ).
fof(cc2_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  ( ~ (v7_struct_0(A))  =>  ~ (v2_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(cc9_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v13_struct_0(A, k5_ordinal1) => v2_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(rc1_relset_1, axiom,  (! [A, B] :  (? [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) &  (v1_xboole_0(C) &  (v1_relat_1(C) &  (v4_relat_1(C, A) & v5_relat_1(C, B)) ) ) ) ) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(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(free_g1_incsp_1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B)))) )  =>  (! [D, E, F] :  (g1_incsp_1(A, B, C)=g1_incsp_1(D, E, F) =>  (A=D &  (B=E & C=F) ) ) ) ) ) ).
fof(abstractness_v1_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v1_incsp_1(A) => A=g1_incsp_1(u1_incsp_1(A), u2_incsp_1(A), u3_incsp_1(A))) ) ) ).
fof(existence_l1_incsp_1, axiom,  (? [A] : l1_incsp_1(A)) ).
fof(existence_l1_struct_0, axiom,  (? [A] : l1_struct_0(A)) ).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(dt_g1_incsp_1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B)))) )  =>  (v1_incsp_1(g1_incsp_1(A, B, C)) & l1_incsp_1(g1_incsp_1(A, B, C))) ) ) ).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_zfmisc_1, axiom, $true).
fof(dt_l1_incsp_1, axiom, $true).
fof(dt_l1_struct_0, axiom, $true).
fof(dt_m1_subset_1, axiom, $true).
fof(cc1_relset_1, axiom,  (! [A, B] :  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) => v1_relat_1(C)) ) ) ).
fof(cc1_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v2_struct_0(A) => v7_struct_0(A)) ) ) ).
fof(cc2_relset_1, axiom,  (! [A, B] :  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) =>  (v4_relat_1(C, A) & v5_relat_1(C, B)) ) ) ) ).
fof(cc3_relset_1, axiom,  (! [A, B] :  (v1_xboole_0(A) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) => v1_xboole_0(C)) ) ) ) ).
fof(cc4_relset_1, axiom,  (! [A, B] :  (v1_xboole_0(A) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(B, A))) => v1_xboole_0(C)) ) ) ) ).
fof(cc4_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v2_struct_0(A) =>  (v2_struct_0(A) & v8_struct_0(A)) ) ) ) ).
fof(cc8_struct_0, axiom,  (! [A] :  (l1_struct_0(A) =>  (v2_struct_0(A) => v13_struct_0(A, k5_ordinal1)) ) ) ).
fof(fc1_struct_0, axiom,  (! [A] :  ( (v2_struct_0(A) & l1_struct_0(A))  => v1_xboole_0(u1_struct_0(A))) ) ).
fof(fc2_struct_0, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  & l1_struct_0(A))  =>  ~ (v1_xboole_0(u1_struct_0(A))) ) ) ).
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(t2_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, B) =>  (v1_xboole_0(B) | r2_tarski(A, B)) ) ) ) ).
fof(t3_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, k1_zfmisc_1(B)) <=> r1_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)) ) ) ) ) ).
fof(t2_tarski, axiom,  (! [A] :  (! [B] :  ( (! [C] :  (r2_hidden(C, A) <=> r2_hidden(C, B)) )  => A=B) ) ) ).
fof(fraenkel_a_1_0_incproj, axiom,  (! [A, B] :  ( ( ~ (v2_struct_0(B))  &  (v2_collsp(B) &  (v3_collsp(B) &  (v4_collsp(B) & l1_collsp(B)) ) ) )  =>  (r2_hidden(A, a_1_0_incproj(B)) <=>  (? [C] :  (m1_subset_1(C, k1_zfmisc_1(u1_struct_0(B))) &  (A=C & m1_incproj(C, B)) ) ) ) ) ) ).
fof(existence_l1_collsp, axiom,  (? [A] : l1_collsp(A)) ).
fof(dt_k1_incproj, axiom, $true).
fof(dt_k2_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  => m1_subset_1(k2_incproj(A), k1_zfmisc_1(k2_zfmisc_1(u1_struct_0(A), k1_incproj(A))))) ) ).
fof(dt_k3_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  => l1_incsp_1(k3_incproj(A))) ) ).
fof(dt_l1_collsp, axiom,  (! [A] :  (l1_collsp(A) => l1_struct_0(A)) ) ).
fof(dt_u1_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  ~ (v1_xboole_0(u1_incsp_1(A))) ) ) ).
fof(dt_u1_struct_0, axiom, $true).
fof(dt_u2_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  ~ (v1_xboole_0(u2_incsp_1(A))) ) ) ).
fof(dt_u3_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) => m1_subset_1(u3_incsp_1(A), k1_zfmisc_1(k2_zfmisc_1(u1_incsp_1(A), u2_incsp_1(A))))) ) ).
fof(fc1_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  =>  ~ (v1_xboole_0(k1_incproj(A))) ) ) ).
fof(fc2_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  => v1_incsp_1(k3_incproj(A))) ) ).
fof(d1_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  => k1_incproj(A)=a_1_0_incproj(A)) ) ).
fof(d3_incproj, axiom,  (! [A] :  ( ( ~ (v2_struct_0(A))  &  (v2_collsp(A) &  (v3_collsp(A) &  (v4_collsp(A) & l1_collsp(A)) ) ) )  => k3_incproj(A)=g1_incsp_1(u1_struct_0(A), k1_incproj(A), k2_incproj(A))) ) ).
