% Mizar problem: t80_classes2,classes2,1451,40 
fof(t80_classes2,conjecture,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k16_classes2(A),k16_classes2(B)) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,antisymmetry_r2_xboole_0,cc1_classes2,cc1_ordinal1,cc2_classes2,cc2_ordinal1,cc3_classes2,cc3_ordinal1,commutativity_k2_xboole_0,connectedness_r1_ordinal1,d1_ordinal1,d2_classes2,d4_card_3,d4_classes2,d5_funct_1,d8_xboole_0,dt_k13_classes2,dt_k15_classes2,dt_k16_classes2,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_relat_1,dt_k2_xboole_0,dt_k3_card_3,dt_k3_tarski,dt_m1_subset_1,existence_m1_subset_1,fc1_classes1,fc1_classes2,fc1_ordinal1,fc1_subset_1,fc2_ordinal1,fc2_subset_1,fc3_classes2,fc3_ordinal1,fc4_classes2,fc4_ordinal1,fc5_ordinal1,fc6_classes2,fc8_classes2,idempotence_k2_xboole_0,irreflexivity_r2_xboole_0,l81_classes2,rc1_classes2,rc1_ordinal1,rc1_ordinal2,rc1_partfun1,rc1_subset_1,rc2_classes2,rc2_ordinal1,rc2_subset_1,rc3_ordinal1,rc4_ordinal1,redefinition_r1_ordinal1,reflexivity_r1_ordinal1,reflexivity_r1_tarski,s1_ordinal2__e4_74__classes2,s2_ordinal2__e5_74_2__classes2,t1_boole,t1_subset,t1_xboole_1,t21_ordinal1,t26_ordinal1,t2_subset,t34_ordinal1,t3_subset,t41_ordinal1,t4_subset,t59_classes2,t5_classes1,t5_subset,t6_boole,t7_boole,t8_boole,t92_zfmisc_1]),
    [file(classes2,t80_classes2)]).

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

fof(antisymmetry_r2_xboole_0,axiom,(
    ! [A,B] : 
      ( r2_xboole_0(A,B)
     => ~ r2_xboole_0(B,A) ) ),
    file(xboole_0,r2_xboole_0),
    []).

fof(cc1_classes2,axiom,(
    ! [A] : 
      ( v2_classes1(A)
     => v1_classes1(A) ) ),
    file(classes2,cc1_classes2),
    []).

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

fof(cc2_classes2,axiom,(
    ! [A] : 
      ( v1_classes2(A)
     => ( v1_ordinal1(A)
        & v1_classes1(A)
        & v2_classes1(A) ) ) ),
    file(classes2,cc2_classes2),
    []).

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

fof(cc3_classes2,axiom,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_classes1(A) )
     => v1_classes2(A) ) ),
    file(classes2,cc3_classes2),
    []).

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

fof(d2_classes2,axiom,(
    k13_classes2 = k1_classes1(k1_xboole_0) ),
    file(classes2,d2_classes2),
    []).

fof(d4_card_3,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k3_card_3(A) = k3_tarski(k2_relat_1(A)) ) ),
    file(card_3,d4_card_3),
    []).

fof(d4_classes2,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & v1_classes2(B) )
     => ( B = k15_classes2(A)
      <=> ( r1_tarski(A,B)
          & ! [C] : 
              ( ( ~ v1_xboole_0(C)
                & v1_classes2(C) )
             => ( r1_tarski(A,C)
               => r1_tarski(B,C) ) ) ) ) ) ),
    file(classes2,d4_classes2),
    []).

fof(d5_funct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) ),
    file(funct_1,d5_funct_1),
    []).

fof(d8_xboole_0,axiom,(
    ! [A,B] : 
      ( r2_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        & A != B ) ) ),
    file(xboole_0,d8_xboole_0),
    []).

fof(dt_k13_classes2,axiom,
    ( ~ v1_xboole_0(k13_classes2)
    & v1_classes2(k13_classes2) ),
    file(classes2,k13_classes2),
    []).

fof(dt_k15_classes2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k15_classes2(A))
      & v1_classes2(k15_classes2(A)) ) ),
    file(classes2,k15_classes2),
    []).

fof(dt_k16_classes2,axiom,(
    $true ),
    file(classes2,k16_classes2),
    []).

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_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_card_3,axiom,(
    $true ),
    file(card_3,k3_card_3),
    []).

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

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_classes2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_classes1(A))
      & v1_classes1(k1_classes1(A))
      & v2_classes1(k1_classes1(A)) ) ),
    file(classes2,fc1_classes2),
    []).

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_classes2,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( v1_ordinal1(k1_classes1(A))
        & ~ v1_xboole_0(k1_classes1(A))
        & v1_classes1(k1_classes1(A))
        & v2_classes1(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) ),
    file(classes2,fc3_classes2),
    []).

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_classes2,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k1_classes1(A))
        & ~ v1_xboole_0(k1_classes1(A))
        & v1_classes1(k1_classes1(A))
        & v2_classes1(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) ),
    file(classes2,fc4_classes2),
    []).

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(fc6_classes2,axiom,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ( v1_ordinal1(k1_classes1(A))
        & ~ v1_xboole_0(k1_classes1(A))
        & v1_classes1(k1_classes1(A))
        & v2_classes1(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) ),
    file(classes2,fc6_classes2),
    []).

fof(fc8_classes2,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k16_classes2(A))
        & ~ v1_xboole_0(k16_classes2(A))
        & v1_classes1(k16_classes2(A))
        & v2_classes1(k16_classes2(A))
        & v1_classes2(k16_classes2(A)) ) ) ),
    file(classes2,fc8_classes2),
    []).

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

fof(irreflexivity_r2_xboole_0,axiom,(
    ! [A,B] : ~ r2_xboole_0(A,A) ),
    file(xboole_0,r2_xboole_0),
    []).

fof(l81_classes2,axiom,
    ( k16_classes2(k1_xboole_0) = k13_classes2
    & ! [A] : 
        ( v3_ordinal1(A)
       => k16_classes2(k1_ordinal1(A)) = k1_classes1(k16_classes2(A)) )
    & ! [A] : 
        ( v3_ordinal1(A)
       => ! [B] : 
            ( ( v1_relat_1(B)
              & v1_funct_1(B)
              & v5_ordinal1(B) )
           => ( ( v4_ordinal1(A)
                & k1_relat_1(B) = A
                & ! [C] : 
                    ( v3_ordinal1(C)
                   => ( r2_hidden(C,A)
                     => k1_funct_1(B,C) = k16_classes2(C) ) ) )
             => ( A = k1_xboole_0
                | k16_classes2(A) = k15_classes2(k3_card_3(B)) ) ) ) ) ),
    file(classes2,l81_classes2),
    []).

fof(rc1_classes2,axiom,(
    ? [A] : 
      ( v1_ordinal1(A)
      & ~ v1_xboole_0(A)
      & v1_classes1(A)
      & v2_classes1(A)
      & v1_classes2(A) ) ),
    file(classes2,rc1_classes2),
    []).

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_partfun1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    []).

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_classes2,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] : 
          ( m1_subset_1(B,A)
          & ~ v1_xboole_0(B) ) ) ),
    file(classes2,rc2_classes2),
    []).

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_r1_ordinal1,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( r1_ordinal1(A,B)
      <=> r1_tarski(A,B) ) ) ),
    file(ordinal1,r1_ordinal1),
    []).

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(s1_ordinal2__e4_74__classes2,axiom,
    ( ( ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(A,k1_xboole_0)
           => r2_hidden(k16_classes2(A),k16_classes2(k1_xboole_0)) ) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( ! [C] : 
                ( v3_ordinal1(C)
               => ( r2_hidden(C,B)
                 => r2_hidden(k16_classes2(C),k16_classes2(B)) ) )
           => ! [D] : 
                ( v3_ordinal1(D)
               => ( r2_hidden(D,k1_ordinal1(B))
                 => r2_hidden(k16_classes2(D),k16_classes2(k1_ordinal1(B))) ) ) ) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( ( v4_ordinal1(B)
              & ! [E] : 
                  ( v3_ordinal1(E)
                 => ( r2_hidden(E,B)
                   => ! [F] : 
                        ( v3_ordinal1(F)
                       => ( r2_hidden(F,E)
                         => r2_hidden(k16_classes2(F),k16_classes2(E)) ) ) ) ) )
           => ( B = k1_xboole_0
              | ! [G] : 
                  ( v3_ordinal1(G)
                 => ( r2_hidden(G,B)
                   => r2_hidden(k16_classes2(G),k16_classes2(B)) ) ) ) ) ) )
   => ! [B] : 
        ( v3_ordinal1(B)
       => ! [H] : 
            ( v3_ordinal1(H)
           => ( r2_hidden(H,B)
             => r2_hidden(k16_classes2(H),k16_classes2(B)) ) ) ) ),
    file(classes2,s1_ordinal2__e4_74__classes2),
    []).

fof(s2_ordinal2__e5_74_2__classes2,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ? [B] : 
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & v5_ordinal1(B)
          & k1_relat_1(B) = A
          & ! [C] : 
              ( v3_ordinal1(C)
             => ( r2_hidden(C,A)
               => k1_funct_1(B,C) = k16_classes2(C) ) ) ) ) ),
    file(classes2,s2_ordinal2__e5_74_2__classes2),
    []).

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(t21_ordinal1,axiom,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_xboole_0(A,B)
           => r2_hidden(A,B) ) ) ) ),
    file(ordinal1,t21_ordinal1),
    []).

fof(t26_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
            | r2_hidden(B,A) ) ) ) ),
    file(ordinal1,t26_ordinal1),
    []).

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

fof(t34_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,k1_ordinal1(B))
          <=> r1_ordinal1(A,B) ) ) ) ),
    file(ordinal1,t34_ordinal1),
    []).

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

fof(t41_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
      <=> ! [B] : 
            ( v3_ordinal1(B)
           => ( r2_hidden(B,A)
             => r2_hidden(k1_ordinal1(B),A) ) ) ) ) ),
    file(ordinal1,t41_ordinal1),
    []).

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(t59_classes2,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ! [C] : 
              ( ( ~ v1_xboole_0(C)
                & v1_classes2(C) )
             => ( ( r2_hidden(A,B)
                  & r2_hidden(B,C) )
               => r2_hidden(A,C) ) ) ) ) ),
    file(classes2,t59_classes2),
    []).

fof(t5_classes1,axiom,(
    ! [A] : r2_hidden(A,k1_classes1(A)) ),
    file(classes1,t5_classes1),
    []).

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),
    []).

fof(t92_zfmisc_1,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => r1_tarski(A,k3_tarski(B)) ) ),
    file(zfmisc_1,t92_zfmisc_1),
    []).