% Mizar problem: t64_domain_1,domain_1,662,43 
fof(t64_domain_1,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k7_subset_1(A,B,C) = a_3_8_domain_1(A,B,C) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k2_xboole_0,commutativity_k5_xboole_0,commutativity_k7_subset_1,d6_xboole_0,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k4_xboole_0,dt_k5_xboole_0,dt_k7_subset_1,dt_m1_subset_1,existence_m1_subset_1,fc1_subset_1,fc1_xboole_0,fc2_xboole_0,fc3_xboole_0,fraenkel_a_3_6_domain_1,fraenkel_a_3_8_domain_1,fraenkel_a_4_4_domain_1,fraenkel_a_4_6_domain_1,idempotence_k2_xboole_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,redefinition_k7_subset_1,reflexivity_r1_tarski,s6_domain_1__e2_43__domain_1,t1_boole,t1_subset,t2_subset,t2_tarski,t3_boole,t3_subset,t4_boole,t4_subset,t5_boole,t5_subset,t62_domain_1,t6_boole,t7_boole,t8_boole]),
    [file(domain_1,t64_domain_1)]).

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

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(commutativity_k5_xboole_0,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k5_xboole_0(B,A) ),
    file(xboole_0,k5_xboole_0),
    []).

fof(commutativity_k7_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k7_subset_1(A,B,C) = k7_subset_1(A,C,B) ) ),
    file(subset_1,k7_subset_1),
    []).

fof(d6_xboole_0,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) ),
    file(xboole_0,d6_xboole_0),
    []).

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_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    []).

fof(dt_k5_xboole_0,axiom,(
    $true ),
    file(xboole_0,k5_xboole_0),
    []).

fof(dt_k7_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k7_subset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(subset_1,k7_subset_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(fc2_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    []).

fof(fc3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    []).

fof(fraenkel_a_3_6_domain_1,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_6_domain_1(B,C,D))
      <=> ? [E] : 
            ( m1_subset_1(E,B)
            & A = E
            & ( ~ r2_hidden(E,C)
            <=> r2_hidden(E,D) ) ) ) ) ),
    file(domain_1,a_3_6_domain_1),
    []).

fof(fraenkel_a_3_8_domain_1,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_8_domain_1(B,C,D))
      <=> ? [E] : 
            ( m1_subset_1(E,B)
            & A = E
            & ~ ( r2_hidden(E,C)
              <=> r2_hidden(E,D) ) ) ) ) ),
    file(domain_1,a_3_8_domain_1),
    []).

fof(fraenkel_a_4_4_domain_1,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B))
        & ~ v1_xboole_0(E) )
     => ( r2_hidden(A,a_4_4_domain_1(B,C,D,E))
      <=> ? [F] : 
            ( m1_subset_1(F,E)
            & A = F
            & ( ~ r2_hidden(F,C)
            <=> r2_hidden(F,D) ) ) ) ) ),
    file(domain_1,a_4_4_domain_1),
    []).

fof(fraenkel_a_4_6_domain_1,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B))
        & ~ v1_xboole_0(E) )
     => ( r2_hidden(A,a_4_6_domain_1(B,C,D,E))
      <=> ? [F] : 
            ( m1_subset_1(F,E)
            & A = F
            & ~ ( r2_hidden(F,C)
              <=> r2_hidden(F,D) ) ) ) ) ),
    file(domain_1,a_4_6_domain_1),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

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_k7_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k7_subset_1(A,B,C) = k5_xboole_0(B,C) ) ),
    file(subset_1,k7_subset_1),
    []).

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

fof(s6_domain_1__e2_43__domain_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => ! [D] : 
          ( ~ v1_xboole_0(D)
         => ( ! [E] : 
                ( m1_subset_1(E,D)
               => ( ( ~ r2_hidden(E,B)
                  <=> r2_hidden(E,C) )
                <=> ~ ( r2_hidden(E,B)
                    <=> r2_hidden(E,C) ) ) )
           => a_4_4_domain_1(A,B,C,D) = a_4_6_domain_1(A,B,C,D) ) ) ) ),
    file(domain_1,s6_domain_1__e2_43__domain_1),
    []).

fof(t1_boole,axiom,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    []).

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_boole,axiom,(
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t3_boole),
    []).

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

fof(t4_boole,axiom,(
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ),
    file(boole,t4_boole),
    []).

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_boole,axiom,(
    ! [A] : k5_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t5_boole),
    []).

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(t62_domain_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k7_subset_1(A,B,C) = a_3_6_domain_1(A,B,C) ) ) ) ),
    file(domain_1,t62_domain_1),
    []).

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),
    []).