% Mizar problem: t54_domain_1,domain_1,507,59 
fof(t54_domain_1,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ~ v1_xboole_0(C)
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ! [E] : 
                      ( m1_subset_1(E,k1_zfmisc_1(B))
                     => ! [F] : 
                          ( m1_subset_1(F,k1_zfmisc_1(C))
                         => k13_mcart_1(A,B,C,D,E,F) = a_6_0_domain_1(A,B,C,D,E,F) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,d10_xboole_0,d3_tarski,dt_k13_mcart_1,dt_k1_xboole_0,dt_k1_zfmisc_1,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_subset_1,fc1_xboole_0,fc5_subset_1,fraenkel_a_6_0_domain_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,redefinition_k13_mcart_1,redefinition_k4_domain_1,reflexivity_r1_tarski,t1_subset,t2_subset,t2_tarski,t3_subset,t48_mcart_1,t4_subset,t5_subset,t6_boole,t73_mcart_1,t76_mcart_1,t7_boole,t8_boole]),
    [file(domain_1,t54_domain_1)]).

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

fof(d10_xboole_0,axiom,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    []).

fof(d3_tarski,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    []).

fof(dt_k13_mcart_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( m1_subset_1(D,k1_zfmisc_1(A))
        & m1_subset_1(E,k1_zfmisc_1(B))
        & m1_subset_1(F,k1_zfmisc_1(C)) )
     => m1_subset_1(k13_mcart_1(A,B,C,D,E,F),k1_zfmisc_1(k3_zfmisc_1(A,B,C))) ) ),
    file(mcart_1,k13_mcart_1),
    []).

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

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_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_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_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_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(fraenkel_a_6_0_domain_1,axiom,(
    ! [A,B,C,D,E,F,G] : 
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_subset_1(E,k1_zfmisc_1(B))
        & m1_subset_1(F,k1_zfmisc_1(C))
        & m1_subset_1(G,k1_zfmisc_1(D)) )
     => ( r2_hidden(A,a_6_0_domain_1(B,C,D,E,F,G))
      <=> ? [H,I,J] : 
            ( m1_subset_1(H,B)
            & m1_subset_1(I,C)
            & m1_subset_1(J,D)
            & A = k4_domain_1(B,C,D,H,I,J)
            & r2_hidden(H,E)
            & r2_hidden(I,F)
            & r2_hidden(J,G) ) ) ) ),
    file(domain_1,a_6_0_domain_1),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

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

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

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

fof(redefinition_k13_mcart_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( m1_subset_1(D,k1_zfmisc_1(A))
        & m1_subset_1(E,k1_zfmisc_1(B))
        & m1_subset_1(F,k1_zfmisc_1(C)) )
     => k13_mcart_1(A,B,C,D,E,F) = k3_zfmisc_1(D,E,F) ) ),
    file(mcart_1,k13_mcart_1),
    []).

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(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

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(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_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(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

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

fof(t73_mcart_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( r2_hidden(k3_mcart_1(A,B,C),k3_zfmisc_1(D,E,F))
    <=> ( r2_hidden(A,D)
        & r2_hidden(B,E)
        & r2_hidden(C,F) ) ) ),
    file(mcart_1,t73_mcart_1),
    []).

fof(t76_mcart_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k1_zfmisc_1(A))
     => ! [E] : 
          ( m1_subset_1(E,k1_zfmisc_1(B))
         => ! [F] : 
              ( m1_subset_1(F,k1_zfmisc_1(C))
             => ! [G] : 
                  ( m1_subset_1(G,k3_zfmisc_1(A,B,C))
                 => ( r2_hidden(G,k3_zfmisc_1(D,E,F))
                   => ( r2_hidden(k5_mcart_1(A,B,C,G),D)
                      & r2_hidden(k6_mcart_1(A,B,C,G),E)
                      & r2_hidden(k7_mcart_1(A,B,C,G),F) ) ) ) ) ) ) ),
    file(mcart_1,t76_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),
    []).