% Mizar problem: t21_classes1,classes1,596,35 
fof(t21_classes1,conjecture,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( k3_classes1(A,B) = k3_classes1(A,k1_ordinal1(B))
       => k3_classes1(A,B) = k1_classes1(A) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_ordinal1,cc2_ordinal1,cc3_ordinal1,commutativity_k2_xboole_0,commutativity_k3_xboole_0,connectedness_r1_ordinal1,d10_xboole_0,d1_classes1,d1_ordinal1,d1_tarski,d2_classes1,d2_xboole_0,d3_classes1,d4_classes1,dt_k1_classes1,dt_k1_funct_1,dt_k1_ordinal1,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_classes1,dt_k2_relat_1,dt_k2_xboole_0,dt_k3_classes1,dt_k3_tarski,dt_k3_xboole_0,dt_m1_subset_1,existence_m1_subset_1,fc1_classes1,fc1_ordinal1,fc1_subset_1,fc2_ordinal1,fc2_subset_1,fc3_ordinal1,fc4_ordinal1,fc5_ordinal1,fraenkel_a_2_0_classes1,fraenkel_a_2_1_classes1,fraenkel_a_2_4_classes1,fraenkel_a_2_5_classes1,idempotence_k2_xboole_0,idempotence_k3_xboole_0,l15_classes1,rc1_ordinal1,rc1_ordinal2,rc1_subset_1,rc2_ordinal1,rc2_subset_1,rc3_ordinal1,rc4_ordinal1,redefinition_k3_classes1,redefinition_r1_ordinal1,redefinition_r2_wellord2,reflexivity_r1_ordinal1,reflexivity_r1_tarski,reflexivity_r2_wellord2,symmetry_r2_wellord2,t13_classes1,t19_classes1,t1_boole,t1_subset,t1_xboole_1,t2_boole,t2_subset,t2_tarski,t2_xboole_1,t3_subset,t44_card_1,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(classes1,t21_classes1)]).

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

fof(cc1_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,cc1_ordinal1),
    []).

fof(cc2_ordinal1,axiom,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_ordinal1(A) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,cc2_ordinal1),
    []).

fof(cc3_ordinal1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(ordinal1,cc3_ordinal1),
    []).

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_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_xboole_0),
    []).

fof(connectedness_r1_ordinal1,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( r1_ordinal1(A,B)
        | r1_ordinal1(B,A) ) ) ),
    file(ordinal1,r1_ordinal1),
    []).

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

fof(d1_classes1,axiom,(
    ! [A] : 
      ( v1_classes1(A)
    <=> ! [B,C] : 
          ( ( r2_hidden(B,A)
            & r1_tarski(C,B) )
         => r2_hidden(C,A) ) ) ),
    file(classes1,d1_classes1),
    []).

fof(d1_ordinal1,axiom,(
    ! [A] : k1_ordinal1(A) = k2_xboole_0(A,k1_tarski(A)) ),
    file(ordinal1,d1_ordinal1),
    []).

fof(d1_tarski,axiom,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    []).

fof(d2_classes1,axiom,(
    ! [A] : 
      ( v2_classes1(A)
    <=> ( v1_classes1(A)
        & ! [B] : 
            ( r2_hidden(B,A)
           => r2_hidden(k1_zfmisc_1(B),A) )
        & ! [B] : ~ ( r1_tarski(B,A)
            & ~ r2_tarski(B,A)
            & ~ r2_hidden(B,A) ) ) ) ),
    file(classes1,d2_classes1),
    []).

fof(d2_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    []).

fof(d3_classes1,axiom,(
    ! [A,B] : 
      ( r1_classes1(A,B)
    <=> ( r2_hidden(A,B)
        & v2_classes1(B) ) ) ),
    file(classes1,d3_classes1),
    []).

fof(d4_classes1,axiom,(
    ! [A,B] : 
      ( B = k1_classes1(A)
    <=> ( r1_classes1(A,B)
        & ! [C] : 
            ( r1_classes1(A,C)
           => r1_tarski(B,C) ) ) ) ),
    file(classes1,d4_classes1),
    []).

fof(dt_k1_classes1,axiom,(
    $true ),
    file(classes1,k1_classes1),
    []).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    []).

fof(dt_k1_ordinal1,axiom,(
    $true ),
    file(ordinal1,k1_ordinal1),
    []).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    []).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    []).

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_classes1,axiom,(
    $true ),
    file(classes1,k2_classes1),
    []).

fof(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    []).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k3_classes1,axiom,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => m1_subset_1(k3_classes1(A,B),k1_zfmisc_1(k1_classes1(A))) ) ),
    file(classes1,k3_classes1),
    []).

fof(dt_k3_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    []).

fof(dt_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    []).

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_classes1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_classes1(A)) ),
    file(classes1,fc1_classes1),
    []).

fof(fc1_ordinal1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_ordinal1(A)) ),
    file(ordinal1,fc1_ordinal1),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc2_ordinal1,axiom,
    ( v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_xboole_0(k1_xboole_0)
    & v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0)
    & v3_ordinal1(k1_xboole_0) ),
    file(ordinal1,fc2_ordinal1),
    []).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    []).

fof(fc3_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ~ v1_xboole_0(k1_ordinal1(A))
        & v1_ordinal1(k1_ordinal1(A))
        & v2_ordinal1(k1_ordinal1(A))
        & v3_ordinal1(k1_ordinal1(A)) ) ) ),
    file(ordinal1,fc3_ordinal1),
    []).

fof(fc4_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k3_tarski(A))
        & v2_ordinal1(k3_tarski(A))
        & v3_ordinal1(k3_tarski(A)) ) ) ),
    file(ordinal1,fc4_ordinal1),
    []).

fof(fc5_ordinal1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ( v1_ordinal1(k1_relat_1(A))
        & v2_ordinal1(k1_relat_1(A))
        & v3_ordinal1(k1_relat_1(A)) ) ) ),
    file(ordinal1,fc5_ordinal1),
    []).

fof(fraenkel_a_2_0_classes1,axiom,(
    ! [A,B,C] : 
      ( v3_ordinal1(C)
     => ( r2_hidden(A,a_2_0_classes1(B,C))
      <=> ? [D] : 
            ( m1_subset_1(D,k1_classes1(B))
            & A = D
            & ? [E] : 
                ( m1_subset_1(E,k1_classes1(B))
                & r2_hidden(E,k2_classes1(B,C))
                & r1_tarski(D,E) ) ) ) ) ),
    file(classes1,a_2_0_classes1),
    []).

fof(fraenkel_a_2_1_classes1,axiom,(
    ! [A,B,C] : 
      ( v3_ordinal1(C)
     => ( r2_hidden(A,a_2_1_classes1(B,C))
      <=> ? [D] : 
            ( m1_subset_1(D,k1_classes1(B))
            & A = k1_zfmisc_1(D)
            & r2_hidden(D,k2_classes1(B,C)) ) ) ) ),
    file(classes1,a_2_1_classes1),
    []).

fof(fraenkel_a_2_4_classes1,axiom,(
    ! [A,B,C] : 
      ( v3_ordinal1(C)
     => ( r2_hidden(A,a_2_4_classes1(B,C))
      <=> ? [D] : 
            ( m1_subset_1(D,k1_classes1(B))
            & A = D
            & ? [E] : 
                ( m1_subset_1(E,k1_classes1(B))
                & r2_hidden(E,k3_classes1(B,C))
                & r1_tarski(D,E) ) ) ) ) ),
    file(classes1,a_2_4_classes1),
    []).

fof(fraenkel_a_2_5_classes1,axiom,(
    ! [A,B,C] : 
      ( v3_ordinal1(C)
     => ( r2_hidden(A,a_2_5_classes1(B,C))
      <=> ? [D] : 
            ( m1_subset_1(D,k1_classes1(B))
            & A = k1_zfmisc_1(D)
            & r2_hidden(D,k3_classes1(B,C)) ) ) ) ),
    file(classes1,a_2_5_classes1),
    []).

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

fof(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    []).

fof(l15_classes1,axiom,(
    ! [A] : 
      ( k2_classes1(A,k1_xboole_0) = k1_tarski(A)
      & ! [B] : 
          ( v3_ordinal1(B)
         => k2_classes1(A,k1_ordinal1(B)) = k2_xboole_0(k2_xboole_0(a_2_0_classes1(A,B),a_2_1_classes1(A,B)),k3_xboole_0(k1_zfmisc_1(k2_classes1(A,B)),k1_classes1(A))) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C) )
             => ( ( v4_ordinal1(B)
                  & k1_relat_1(C) = B
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,B)
                       => k1_funct_1(C,D) = k2_classes1(A,D) ) ) )
               => ( B = k1_xboole_0
                  | k2_classes1(A,B) = k3_xboole_0(k3_tarski(k2_relat_1(C)),k1_classes1(A)) ) ) ) ) ) ),
    file(classes1,l15_classes1),
    []).

fof(rc1_ordinal1,axiom,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc1_ordinal1),
    []).

fof(rc1_ordinal2,axiom,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal1(A) ) ),
    file(ordinal2,rc1_ordinal2),
    []).

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(rc2_ordinal1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc2_ordinal1),
    []).

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(rc3_ordinal1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc3_ordinal1),
    []).

fof(rc4_ordinal1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A) ) ),
    file(ordinal1,rc4_ordinal1),
    []).

fof(redefinition_k3_classes1,axiom,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => k3_classes1(A,B) = k2_classes1(A,B) ) ),
    file(classes1,k3_classes1),
    []).

fof(redefinition_r1_ordinal1,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( r1_ordinal1(A,B)
      <=> r1_tarski(A,B) ) ) ),
    file(ordinal1,r1_ordinal1),
    []).

fof(redefinition_r2_wellord2,axiom,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
    <=> r2_tarski(A,B) ) ),
    file(wellord2,r2_wellord2),
    []).

fof(reflexivity_r1_ordinal1,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => r1_ordinal1(A,A) ) ),
    file(ordinal1,r1_ordinal1),
    []).

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

fof(reflexivity_r2_wellord2,axiom,(
    ! [A,B] : r2_wellord2(A,A) ),
    file(wellord2,r2_wellord2),
    []).

fof(symmetry_r2_wellord2,axiom,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
     => r2_wellord2(B,A) ) ),
    file(wellord2,r2_wellord2),
    []).

fof(t13_classes1,axiom,(
    ! [A,B,C] : 
      ( v3_ordinal1(C)
     => ( r2_hidden(A,k3_classes1(B,k1_ordinal1(C)))
      <=> ~ ( ~ ( r1_tarski(A,k3_classes1(B,C))
                & r2_hidden(A,k1_classes1(B)) )
            & ! [D] : ~ ( r2_hidden(D,k3_classes1(B,C))
                & ( r1_tarski(A,D)
                  | A = k1_zfmisc_1(D) ) ) ) ) ) ),
    file(classes1,t13_classes1),
    []).

fof(t19_classes1,axiom,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ! [C] : 
          ( v3_ordinal1(C)
         => ( r1_ordinal1(B,C)
           => r1_tarski(k3_classes1(A,B),k3_classes1(A,C)) ) ) ) ),
    file(classes1,t19_classes1),
    []).

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(t1_xboole_1,axiom,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    []).

fof(t2_boole,axiom,(
    ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ),
    file(boole,t2_boole),
    []).

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(t2_xboole_1,axiom,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_1),
    []).

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

fof(t44_card_1,axiom,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C)
        & r2_wellord2(A,C) )
     => ( r2_wellord2(A,B)
        & r2_wellord2(B,C) ) ) ),
    file(card_1,t44_card_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(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),
    []).