% Mizar problem: t26_classes2,classes2,460,68 
fof(t26_classes2,conjecture,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( ( v2_classes1(B)
            & r2_hidden(A,k2_ordinal2(B)) )
         => ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(B))
            & r2_hidden(k4_classes1(A),B) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_card_1,cc1_classes2,cc1_ordinal1,cc2_ordinal1,cc3_ordinal1,commutativity_k2_xboole_0,d1_ordinal1,dt_k1_card_1,dt_k1_ordinal1,dt_k1_tarski,dt_k1_xboole_0,dt_k2_ordinal2,dt_k2_xboole_0,dt_k4_classes1,dt_m1_subset_1,existence_m1_subset_1,fc1_ordinal1,fc2_ordinal1,fc2_subset_1,fc3_ordinal1,idempotence_k2_xboole_0,l27_classes2,l28_classes2,l29_classes2,rc1_card_1,rc1_ordinal1,rc1_ordinal2,rc1_partfun1,rc2_ordinal1,rc3_ordinal1,s1_ordinal2__e1_23__classes2,t1_boole,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole]),
    [file(classes2,t26_classes2)]).

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

fof(cc1_card_1,axiom,(
    ! [A] : 
      ( v1_card_1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(card_1,cc1_card_1),
    []).

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

fof(dt_k1_card_1,axiom,(
    ! [A] : v1_card_1(k1_card_1(A)) ),
    file(card_1,k1_card_1),
    []).

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

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_k2_ordinal2,axiom,(
    $true ),
    file(ordinal2,k2_ordinal2),
    []).

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

fof(dt_k4_classes1,axiom,(
    $true ),
    file(classes1,k4_classes1),
    []).

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_ordinal1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_ordinal1(A)) ),
    file(ordinal1,fc1_ordinal1),
    []).

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

fof(l27_classes2,axiom,(
    ! [A] : 
      ( ( v2_classes1(A)
        & r2_hidden(k1_xboole_0,k2_ordinal2(A)) )
     => ( r2_hidden(k1_card_1(k4_classes1(k1_xboole_0)),k1_card_1(A))
        & r2_hidden(k4_classes1(k1_xboole_0),A) ) ) ),
    file(classes2,l27_classes2),
    []).

fof(l28_classes2,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ! [B] : 
            ( ( v2_classes1(B)
              & r2_hidden(A,k2_ordinal2(B)) )
           => ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(B))
              & r2_hidden(k4_classes1(A),B) ) )
       => ! [B] : 
            ( ( v2_classes1(B)
              & r2_hidden(k1_ordinal1(A),k2_ordinal2(B)) )
           => ( r2_hidden(k1_card_1(k4_classes1(k1_ordinal1(A))),k1_card_1(B))
              & r2_hidden(k4_classes1(k1_ordinal1(A)),B) ) ) ) ) ),
    file(classes2,l28_classes2),
    []).

fof(l29_classes2,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ( v4_ordinal1(A)
          & ! [B] : 
              ( v3_ordinal1(B)
             => ( r2_hidden(B,A)
               => ! [C] : 
                    ( ( v2_classes1(C)
                      & r2_hidden(B,k2_ordinal2(C)) )
                   => ( r2_hidden(k1_card_1(k4_classes1(B)),k1_card_1(C))
                      & r2_hidden(k4_classes1(B),C) ) ) ) ) )
       => ( A = k1_xboole_0
          | ! [B] : 
              ( ( v2_classes1(B)
                & r2_hidden(A,k2_ordinal2(B)) )
             => ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(B))
                & r2_hidden(k4_classes1(A),B) ) ) ) ) ) ),
    file(classes2,l29_classes2),
    []).

fof(rc1_card_1,axiom,(
    ? [A] : v1_card_1(A) ),
    file(card_1,rc1_card_1),
    []).

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

fof(s1_ordinal2__e1_23__classes2,axiom,
    ( ( ! [A] : 
          ( ( v2_classes1(A)
            & r2_hidden(k1_xboole_0,k2_ordinal2(A)) )
         => ( r2_hidden(k1_card_1(k4_classes1(k1_xboole_0)),k1_card_1(A))
            & r2_hidden(k4_classes1(k1_xboole_0),A) ) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( ! [C] : 
                ( ( v2_classes1(C)
                  & r2_hidden(B,k2_ordinal2(C)) )
               => ( r2_hidden(k1_card_1(k4_classes1(B)),k1_card_1(C))
                  & r2_hidden(k4_classes1(B),C) ) )
           => ! [D] : 
                ( ( v2_classes1(D)
                  & r2_hidden(k1_ordinal1(B),k2_ordinal2(D)) )
               => ( r2_hidden(k1_card_1(k4_classes1(k1_ordinal1(B))),k1_card_1(D))
                  & r2_hidden(k4_classes1(k1_ordinal1(B)),D) ) ) ) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( ( v4_ordinal1(B)
              & ! [E] : 
                  ( v3_ordinal1(E)
                 => ( r2_hidden(E,B)
                   => ! [F] : 
                        ( ( v2_classes1(F)
                          & r2_hidden(E,k2_ordinal2(F)) )
                       => ( r2_hidden(k1_card_1(k4_classes1(E)),k1_card_1(F))
                          & r2_hidden(k4_classes1(E),F) ) ) ) ) )
           => ( B = k1_xboole_0
              | ! [G] : 
                  ( ( v2_classes1(G)
                    & r2_hidden(B,k2_ordinal2(G)) )
                 => ( r2_hidden(k1_card_1(k4_classes1(B)),k1_card_1(G))
                    & r2_hidden(k4_classes1(B),G) ) ) ) ) ) )
   => ! [B] : 
        ( v3_ordinal1(B)
       => ! [H] : 
            ( ( v2_classes1(H)
              & r2_hidden(B,k2_ordinal2(H)) )
           => ( r2_hidden(k1_card_1(k4_classes1(B)),k1_card_1(H))
              & r2_hidden(k4_classes1(B),H) ) ) ) ),
    file(classes2,s1_ordinal2__e1_23__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(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),
    []).