% Mizar problem: t49_interva1,interva1,1435,7 
fof(t49_interva1, conjecture,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  ~ ( (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  => k10_interva1(A, k3_interva1(A, B, B))=k2_interva1(A, k1_subset_1(A), k1_subset_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(cc1_subset_1, axiom,  (! [A] :  (v1_xboole_0(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_xboole_0(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_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_subset_1, axiom,  (! [A] :  (v1_xboole_0(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  ~ (v1_subset_1(B, A)) ) ) ) ) ).
fof(commutativity_k3_setfam_1, axiom,  (! [A, B] : k3_setfam_1(A, B)=k3_setfam_1(B, A)) ).
fof(commutativity_k3_xboole_0, axiom,  (! [A, B] : k3_xboole_0(A, B)=k3_xboole_0(B, A)) ).
fof(commutativity_k9_subset_1, axiom,  (! [A, B, C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => k9_subset_1(A, B, C)=k9_subset_1(A, C, B)) ) ).
fof(d10_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  => k10_interva1(A, B)=k9_interva1(A, k2_interva1(A, k2_subset_1(A), k2_subset_1(A)), B)) ) ) ) ).
fof(d1_interva1, axiom,  (! [A] :  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => k1_interva1(A, B, C)=a_3_0_interva1(A, B, C)) ) ) ) ) ).
fof(d2_subset_1, axiom,  (! [A] : k1_subset_1(A)=k1_xboole_0) ).
fof(d3_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  =>  (! [C] :  ( ( ~ (v1_xboole_0(C))  & m1_interva1(C, A))  => k3_interva1(A, B, C)=k3_setfam_1(B, C)) ) ) ) ) ) ).
fof(d3_subset_1, axiom,  (! [A] : k2_subset_1(A)=A) ).
fof(d3_tarski, axiom,  (! [A] :  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ).
fof(d4_subset_1, axiom,  (! [A] :  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => k3_subset_1(A, B)=k4_xboole_0(A, B)) ) ) ).
fof(d9_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  =>  (! [C] :  ( ( ~ (v1_xboole_0(C))  & m1_interva1(C, A))  => k9_interva1(A, B, C)=k4_setfam_1(B, C)) ) ) ) ) ) ).
fof(dt_k10_interva1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_interva1(B, A)) )  =>  ( ~ (v1_xboole_0(k10_interva1(A, B)))  & m1_interva1(k10_interva1(A, B), A)) ) ) ).
fof(dt_k1_interva1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => m1_subset_1(k1_interva1(A, B, C), k1_zfmisc_1(k1_zfmisc_1(A)))) ) ).
fof(dt_k1_subset_1, axiom,  (! [A] : m1_subset_1(k1_subset_1(A), k1_zfmisc_1(A))) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_interva1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => m1_interva1(k2_interva1(A, B, C), A)) ) ).
fof(dt_k2_subset_1, axiom,  (! [A] : m1_subset_1(k2_subset_1(A), k1_zfmisc_1(A))) ).
fof(dt_k3_interva1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  &  ( ~ (v1_xboole_0(C))  & m1_interva1(C, A)) ) )  => m1_interva1(k3_interva1(A, B, C), A)) ) ).
fof(dt_k3_setfam_1, axiom, $true).
fof(dt_k3_subset_1, axiom,  (! [A, B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => m1_subset_1(k3_subset_1(A, B), k1_zfmisc_1(A))) ) ).
fof(dt_k3_xboole_0, axiom, $true).
fof(dt_k4_setfam_1, axiom, $true).
fof(dt_k4_xboole_0, axiom, $true).
fof(dt_k5_interva1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_interva1(B, A)) )  => m1_subset_1(k5_interva1(A, B), k1_zfmisc_1(A))) ) ).
fof(dt_k6_interva1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_interva1(B, A)) )  => m1_subset_1(k6_interva1(A, B), k1_zfmisc_1(A))) ) ).
fof(dt_k9_interva1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  &  ( ~ (v1_xboole_0(C))  & m1_interva1(C, A)) ) )  => m1_interva1(k9_interva1(A, B, C), A)) ) ).
fof(dt_k9_subset_1, axiom,  (! [A, B, C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => m1_subset_1(k9_subset_1(A, B, C), k1_zfmisc_1(A))) ) ).
fof(dt_m1_interva1, axiom,  (! [A] :  (! [B] :  (m1_interva1(B, A) => m1_subset_1(B, k1_zfmisc_1(k1_zfmisc_1(A)))) ) ) ).
fof(dt_m1_subset_1, axiom, $true).
fof(existence_m1_interva1, axiom,  (! [A] :  (? [B] : m1_interva1(B, A)) ) ).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(fc13_subset_1, axiom,  (! [A] : v1_xboole_0(k1_subset_1(A))) ).
fof(fc14_subset_1, axiom,  (! [A] :  ~ (v1_subset_1(k2_subset_1(A), A)) ) ).
fof(fc1_interva1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  &  ( ~ (v1_xboole_0(C))  & m1_interva1(C, A)) ) )  =>  ~ (v1_xboole_0(k3_interva1(A, B, C))) ) ) ).
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(fc3_interva1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  &  ( ~ (v1_xboole_0(C))  & m1_interva1(C, A)) ) )  =>  ~ (v1_xboole_0(k9_interva1(A, B, C))) ) ) ).
fof(fc4_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  ~ (v1_xboole_0(k1_interva1(A, k2_subset_1(A), k2_subset_1(A)))) ) ) ).
fof(fc5_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  ~ (v1_xboole_0(k1_interva1(A, k1_subset_1(A), k1_subset_1(A)))) ) ) ).
fof(fraenkel_a_3_0_interva1, axiom,  (! [A, B, C, D] :  ( (m1_subset_1(C, k1_zfmisc_1(B)) & m1_subset_1(D, k1_zfmisc_1(B)))  =>  (r2_hidden(A, a_3_0_interva1(B, C, D)) <=>  (? [E] :  (m1_subset_1(E, k1_zfmisc_1(B)) &  (A=E &  (r1_tarski(C, E) & r1_tarski(E, D)) ) ) ) ) ) ) ).
fof(idempotence_k3_xboole_0, axiom,  (! [A, B] : k3_xboole_0(A, A)=A) ).
fof(idempotence_k9_subset_1, axiom,  (! [A, B, C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => k9_subset_1(A, B, B)=B) ) ).
fof(involutiveness_k3_subset_1, axiom,  (! [A, B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => k3_subset_1(A, k3_subset_1(A, B))=B) ) ).
fof(rc1_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (? [B] :  (m1_interva1(B, A) &  ~ (v1_xboole_0(B)) ) ) ) ) ).
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_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_k2_interva1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => k2_interva1(A, B, C)=k1_interva1(A, B, C)) ) ).
fof(redefinition_k9_subset_1, axiom,  (! [A, B, C] :  (m1_subset_1(C, k1_zfmisc_1(A)) => k9_subset_1(A, B, C)=k3_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(t15_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  => B=k2_interva1(A, k5_interva1(A, B), k6_interva1(A, B))) ) ) ) ).
fof(t18_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  =>  (! [C] :  ( ( ~ (v1_xboole_0(C))  & m1_interva1(C, A))  => k3_interva1(A, B, C)=k2_interva1(A, k9_subset_1(A, k5_interva1(A, B), k5_interva1(A, C)), k9_subset_1(A, k6_interva1(A, B), k6_interva1(A, C)))) ) ) ) ) ) ).
fof(t1_interva1, axiom,  (! [A] :  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(A)) =>  (! [D] :  (r2_tarski(D, k1_interva1(A, B, C)) <=>  (r1_tarski(B, D) & r1_tarski(D, C)) ) ) ) ) ) ) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t2_boole, axiom,  (! [A] : k3_xboole_0(A, k1_xboole_0)=k1_xboole_0) ).
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_boole, axiom,  (! [A] : k4_xboole_0(A, k1_xboole_0)=A) ).
fof(t3_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, k1_zfmisc_1(B)) <=> r1_tarski(A, B)) ) ) ).
fof(t46_interva1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  ( ( ~ (v1_xboole_0(B))  & m1_interva1(B, A))  =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(A)) =>  (! [D] :  (m1_subset_1(D, k1_zfmisc_1(A)) =>  ( (B=k2_interva1(A, C, D) & r1_tarski(C, D))  => k10_interva1(A, B)=k2_interva1(A, k3_subset_1(A, D), k3_subset_1(A, C))) ) ) ) ) ) ) ) ) ).
fof(t4_boole, axiom,  (! [A] : k4_xboole_0(k1_xboole_0, A)=k1_xboole_0) ).
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(t6_interva1, axiom,  (! [A] :  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(A)) =>  (! [D] :  (m1_subset_1(D, k1_zfmisc_1(A)) =>  (! [E] :  (m1_subset_1(E, k1_zfmisc_1(A)) =>  (k1_interva1(A, B, C)=k1_interva1(A, D, E) =>  (k1_interva1(A, B, C)=k1_xboole_0 |  (B=D & C=E) ) ) ) ) ) ) ) ) ) ) ) ).
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)) ) ) ) ) ).
