% Mizar problem: t70_xxreal_1,xxreal_1,1130,5 
fof(t70_xxreal_1, conjecture,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  (! [D] :  (v1_xxreal_0(D) =>  ~ ( (r1_xxreal_0(C, D) & k1_xxreal_1(C, D)=k3_xxreal_1(A, B)) ) ) ) ) ) ) ) ) ) ).
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_membered, axiom,  (! [A] :  (v5_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_int_1(B)) ) ) ) ).
fof(cc11_membered, axiom,  (! [A] :  (v6_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v7_ordinal1(B)) ) ) ) ).
fof(cc12_membered, axiom,  (! [A] :  (v1_xboole_0(A) => v6_membered(A)) ) ).
fof(cc13_membered, axiom,  (! [A] :  (v1_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_membered(B)) ) ) ) ).
fof(cc14_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v2_membered(B)) ) ) ) ).
fof(cc15_membered, axiom,  (! [A] :  (v3_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v3_membered(B)) ) ) ) ).
fof(cc16_membered, axiom,  (! [A] :  (v4_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v4_membered(B)) ) ) ) ).
fof(cc17_membered, axiom,  (! [A] :  (v5_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v5_membered(B)) ) ) ) ).
fof(cc18_membered, axiom,  (! [A] :  (v6_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v6_membered(B)) ) ) ) ).
fof(cc19_membered, axiom,  (! [A] :  (v1_xboole_0(A) => v7_membered(A)) ) ).
fof(cc1_membered, axiom,  (! [A] :  (v6_membered(A) => v5_membered(A)) ) ).
fof(cc1_xxreal_0, axiom,  (! [A] :  (m1_subset_1(A, k6_numbers) => v1_xxreal_0(A)) ) ).
fof(cc2_membered, axiom,  (! [A] :  (v5_membered(A) => v4_membered(A)) ) ).
fof(cc2_xreal_0, axiom,  (! [A] :  (v7_ordinal1(A) => v1_xreal_0(A)) ) ).
fof(cc2_xxreal_0, axiom,  (! [A] :  (v7_ordinal1(A) => v1_xxreal_0(A)) ) ).
fof(cc3_membered, axiom,  (! [A] :  (v4_membered(A) => v3_membered(A)) ) ).
fof(cc3_xreal_0, axiom,  (! [A] :  (v1_xreal_0(A) => v1_xcmplx_0(A)) ) ).
fof(cc4_membered, axiom,  (! [A] :  (v3_membered(A) => v2_membered(A)) ) ).
fof(cc4_xreal_0, axiom,  (! [A] :  (v1_xreal_0(A) => v1_xxreal_0(A)) ) ).
fof(cc5_membered, axiom,  (! [A] :  (v3_membered(A) => v1_membered(A)) ) ).
fof(cc6_membered, axiom,  (! [A] :  (v1_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xcmplx_0(B)) ) ) ) ).
fof(cc7_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xxreal_0(B)) ) ) ) ).
fof(cc8_membered, axiom,  (! [A] :  (v3_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xreal_0(B)) ) ) ) ).
fof(cc9_membered, axiom,  (! [A] :  (v4_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_rat_1(B)) ) ) ) ).
fof(connectedness_r1_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  =>  (r1_xxreal_0(A, B) | r1_xxreal_0(B, A)) ) ) ).
fof(d10_membered, axiom,  (! [A] :  (v4_membered(A) =>  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (v1_rat_1(C) =>  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ) ) ).
fof(d11_membered, axiom,  (! [A] :  (v5_membered(A) =>  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (v1_int_1(C) =>  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ) ) ).
fof(d12_membered, axiom,  (! [A] :  (v6_membered(A) =>  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (v7_ordinal1(C) =>  (r2_tarski(C, A) => r2_tarski(C, B)) ) ) ) ) ) ) ).
fof(d1_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) => k1_xxreal_1(A, B)=a_2_0_xxreal_1(A, B)) ) ) ) ).
fof(d3_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) => k3_xxreal_1(A, B)=a_2_2_xxreal_1(A, B)) ) ) ) ).
fof(d7_membered, axiom,  (! [A] :  (v1_membered(A) =>  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (v1_xcmplx_0(C) =>  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ) ) ).
fof(d8_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (v1_xxreal_0(C) =>  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ) ) ).
fof(d9_membered, axiom,  (! [A] :  (v3_membered(A) =>  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (v1_xreal_0(C) =>  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ) ) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_xxreal_1, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k3_xxreal_1, axiom, $true).
fof(dt_k6_numbers, axiom, $true).
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(fc1_xxreal_1, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v2_membered(k1_xxreal_1(A, B))) ) ).
fof(fc2_membered, axiom, v2_membered(k6_numbers)).
fof(fc3_xxreal_1, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v2_membered(k3_xxreal_1(A, B))) ) ).
fof(fc5_numbers, axiom,  ~ (v1_xboole_0(k6_numbers)) ).
fof(fc5_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  ~ (v1_xboole_0(k1_xxreal_1(A, A))) ) ) ).
fof(fc7_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) => v1_xboole_0(k3_xxreal_1(A, A))) ) ).
fof(fraenkel_a_2_0_xxreal_1, axiom,  (! [A, B, C] :  ( (v1_xxreal_0(B) & v1_xxreal_0(C))  =>  (r2_hidden(A, a_2_0_xxreal_1(B, C)) <=>  (? [D] :  (m1_subset_1(D, k6_numbers) &  (A=D &  (r1_xxreal_0(B, D) & r1_xxreal_0(D, C)) ) ) ) ) ) ) ).
fof(fraenkel_a_2_2_xxreal_1, axiom,  (! [A, B, C] :  ( (v1_xxreal_0(B) & v1_xxreal_0(C))  =>  (r2_hidden(A, a_2_2_xxreal_1(B, C)) <=>  (? [D] :  (m1_subset_1(D, k6_numbers) &  (A=D &  ( ~ (r1_xxreal_0(D, B))  & r1_xxreal_0(D, C)) ) ) ) ) ) ) ).
fof(rc1_membered, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v6_membered(A)) ) ).
fof(rc1_xboole_0, axiom,  (? [A] : v1_xboole_0(A)) ).
fof(rc1_xreal_0, axiom,  (? [A] : v1_xreal_0(A)) ).
fof(rc1_xxreal_0, axiom,  (? [A] : v1_xxreal_0(A)) ).
fof(rc2_membered, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v6_membered(A)) ) ).
fof(rc2_xboole_0, axiom,  (? [A] :  ~ (v1_xboole_0(A)) ) ).
fof(rc2_xreal_0, axiom,  (? [A] : v1_xreal_0(A)) ).
fof(rc2_xxreal_0, axiom,  (? [A] : v1_xxreal_0(A)) ).
fof(rc3_membered, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  &  (v6_membered(A) & v7_membered(A)) ) ) ).
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(reflexivity_r1_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => r1_xxreal_0(A, A)) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t1_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  (r2_hidden(C, k1_xxreal_1(A, B)) <=>  (r1_xxreal_0(A, C) & r1_xxreal_0(C, 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(t2_xxreal_0, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  ( (r1_xxreal_0(A, B) & r1_xxreal_0(B, C))  => r1_xxreal_0(A, C)) ) ) ) ) ) ) ).
fof(t2_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  (r2_hidden(C, k3_xxreal_1(A, B)) <=>  ( ~ (r1_xxreal_0(C, A))  & r1_xxreal_0(C, 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(t53_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  (! [D] :  (v1_xxreal_0(D) =>  (r1_tarski(k3_xxreal_1(C, D), k1_xxreal_1(A, B)) =>  (r1_xxreal_0(D, C) |  (r1_xxreal_0(A, C) & r1_xxreal_0(D, B)) ) ) ) ) ) ) ) ) ) ) ).
fof(t58_xxreal_1, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  (! [D] :  (v1_xxreal_0(D) =>  ( (r1_xxreal_0(C, D) & r1_tarski(k1_xxreal_1(C, D), k3_xxreal_1(A, B)))  =>  ( ~ (r1_xxreal_0(C, A))  & r1_xxreal_0(D, B)) ) ) ) ) ) ) ) ) ) ).
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)) ) ) ) ) ).
