% Mizar problem: t17_domain_1,domain_1,138,54 
fof(t17_domain_1,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ~ v1_xboole_0(C)
             => ! [D] : 
                  ( ~ v1_xboole_0(D)
                 => ( A = k3_zfmisc_1(B,C,D)
                  <=> ! [E] : 
                        ( r2_hidden(E,A)
                      <=> ? [F] : 
                            ( m1_subset_1(F,B)
                            & ? [G] : 
                                ( m1_subset_1(G,C)
                                & ? [H] : 
                                    ( m1_subset_1(H,D)
                                    & E = k3_mcart_1(F,G,H) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[dt_k1_xboole_0,fc1_xboole_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k3_mcart_1,dt_k3_zfmisc_1,dt_m1_subset_1,fc5_subset_1,rc1_xboole_0,rc2_xboole_0,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,t15_domain_1,t16_domain_1]),
    [file(domain_1,t17_domain_1)]).

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

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

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

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

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_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

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

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

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

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),
    []).

fof(t15_domain_1,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( ~ v1_xboole_0(C)
         => ! [D] : 
              ( ~ v1_xboole_0(D)
             => ( r2_hidden(A,k3_zfmisc_1(B,C,D))
              <=> ? [E] : 
                    ( m1_subset_1(E,B)
                    & ? [F] : 
                        ( m1_subset_1(F,C)
                        & ? [G] : 
                            ( m1_subset_1(G,D)
                            & A = k3_mcart_1(E,F,G) ) ) ) ) ) ) ) ),
    file(domain_1,t15_domain_1),
    []).

fof(t16_domain_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ~ v1_xboole_0(C)
             => ! [D] : 
                  ( ~ v1_xboole_0(D)
                 => ( ! [E] : 
                        ( r2_hidden(E,A)
                      <=> ? [F] : 
                            ( m1_subset_1(F,B)
                            & ? [G] : 
                                ( m1_subset_1(G,C)
                                & ? [H] : 
                                    ( m1_subset_1(H,D)
                                    & E = k3_mcart_1(F,G,H) ) ) ) )
                   => A = k3_zfmisc_1(B,C,D) ) ) ) ) ) ),
    file(domain_1,t16_domain_1),
    []).