% Mizar problem: t26_domain_1,domain_1,178,53 
fof(t26_domain_1,conjecture,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( ~ v1_xboole_0(C)
         => ! [D] : 
              ( ~ v1_xboole_0(D)
             => ! [E] : 
                  ( m1_subset_1(E,k3_zfmisc_1(B,C,D))
                 => ( A = k7_mcart_1(B,C,D,E)
                  <=> ! [F] : 
                        ( m1_subset_1(F,B)
                       => ! [G] : 
                            ( m1_subset_1(G,C)
                           => ! [H] : 
                                ( m1_subset_1(H,D)
                               => ( E = k4_domain_1(B,C,D,F,G,H)
                                 => A = H ) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k3_mcart_1,dt_k3_zfmisc_1,dt_k4_domain_1,dt_k5_mcart_1,dt_k6_mcart_1,dt_k7_mcart_1,dt_m1_subset_1,existence_m1_subset_1,fc1_xboole_0,fc5_subset_1,rc1_xboole_0,rc2_xboole_0,redefinition_k4_domain_1,t1_subset,t28_mcart_1,t2_subset,t48_mcart_1,t6_boole,t71_mcart_1,t7_boole,t8_boole]),
    [file(domain_1,t26_domain_1)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k3_mcart_1,axiom,(
    $true ),
    file(mcart_1,k3_mcart_1),
    []).

fof(dt_k3_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k3_zfmisc_1),
    []).

fof(dt_k4_domain_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B)
        & m1_subset_1(F,C) )
     => m1_subset_1(k4_domain_1(A,B,C,D,E,F),k3_zfmisc_1(A,B,C)) ) ),
    file(domain_1,k4_domain_1),
    []).

fof(dt_k5_mcart_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
     => m1_subset_1(k5_mcart_1(A,B,C,D),A) ) ),
    file(mcart_1,k5_mcart_1),
    []).

fof(dt_k6_mcart_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
     => m1_subset_1(k6_mcart_1(A,B,C,D),B) ) ),
    file(mcart_1,k6_mcart_1),
    []).

fof(dt_k7_mcart_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
     => m1_subset_1(k7_mcart_1(A,B,C,D),C) ) ),
    file(mcart_1,k7_mcart_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc5_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C) )
     => ~ v1_xboole_0(k3_zfmisc_1(A,B,C)) ) ),
    file(subset_1,fc5_subset_1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(redefinition_k4_domain_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B)
        & m1_subset_1(F,C) )
     => k4_domain_1(A,B,C,D,E,F) = k3_mcart_1(D,E,F) ) ),
    file(domain_1,k4_domain_1),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t28_mcart_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( k3_mcart_1(A,B,C) = k3_mcart_1(D,E,F)
     => ( A = D
        & B = E
        & C = F ) ) ),
    file(mcart_1,t28_mcart_1),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t48_mcart_1,axiom,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => D = k3_mcart_1(k5_mcart_1(A,B,C,D),k6_mcart_1(A,B,C,D),k7_mcart_1(A,B,C,D)) ) ) ),
    file(mcart_1,t48_mcart_1),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t71_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k3_zfmisc_1(A,B,C))
     => ( ! [F] : 
            ( m1_subset_1(F,A)
           => ! [G] : 
                ( m1_subset_1(G,B)
               => ! [H] : 
                    ( m1_subset_1(H,C)
                   => ( E = k3_mcart_1(F,G,H)
                     => D = H ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k7_mcart_1(A,B,C,E) ) ) ) ),
    file(mcart_1,t71_mcart_1),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).