% Mizar problem: t41_classes1,classes1,1042,40 
fof(t41_classes1,conjecture,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r2_hidden(A,k4_classes1(B))
       => r2_hidden(k3_tarski(A),k4_classes1(B)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,antisymmetry_r2_xboole_0,cc1_ordinal1,cc2_ordinal1,cc3_ordinal1,commutativity_k2_xboole_0,connectedness_r1_ordinal1,d1_ordinal1,d8_xboole_0,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_tarski,dt_k4_classes1,dt_m1_subset_1,existence_m1_subset_1,fc1_ordinal1,fc1_subset_1,fc2_ordinal1,fc2_subset_1,fc3_ordinal1,fc4_ordinal1,fc5_ordinal1,idempotence_k2_xboole_0,irreflexivity_r2_xboole_0,l40_classes1,rc1_ordinal1,rc1_ordinal2,rc1_subset_1,rc2_ordinal1,rc2_subset_1,rc3_ordinal1,rc4_ordinal1,redefinition_r1_ordinal1,reflexivity_r1_ordinal1,reflexivity_r1_tarski,t1_boole,t1_subset,t1_xboole_1,t21_ordinal1,t2_subset,t33_ordinal1,t35_classes1,t36_classes1,t38_classes1,t3_subset,t40_classes1,t42_ordinal1,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,t95_zfmisc_1]),
    [file(classes1,t41_classes1)]).

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_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(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(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_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_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    []).

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(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(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(l40_classes1,axiom,
    ( k4_classes1(k1_xboole_0) = k1_xboole_0
    & ! [A] : 
        ( v3_ordinal1(A)
       => k4_classes1(k1_ordinal1(A)) = k1_zfmisc_1(k4_classes1(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) = k4_classes1(C) ) ) )
             => ( A = k1_xboole_0
                | k4_classes1(A) = k3_tarski(k2_relat_1(B)) ) ) ) ) ),
    file(classes1,l40_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_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(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(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t33_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r1_ordinal1(k1_ordinal1(A),B) ) ) ) ),
    file(ordinal1,t33_ordinal1),
    []).

fof(t35_classes1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
       => ( A = k1_xboole_0
          | ! [B] : 
              ( r2_hidden(B,k4_classes1(A))
            <=> ? [C] : 
                  ( v3_ordinal1(C)
                  & r2_hidden(C,A)
                  & r2_hidden(B,k4_classes1(C)) ) ) ) ) ) ),
    file(classes1,t35_classes1),
    []).

fof(t36_classes1,axiom,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r1_tarski(A,k4_classes1(B))
      <=> r2_hidden(A,k4_classes1(k1_ordinal1(B))) ) ) ),
    file(classes1,t36_classes1),
    []).

fof(t38_classes1,axiom,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r2_hidden(A,k4_classes1(B))
       => r1_tarski(A,k4_classes1(B)) ) ) ),
    file(classes1,t38_classes1),
    []).

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

fof(t40_classes1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r1_tarski(k3_tarski(k4_classes1(A)),k4_classes1(A)) ) ),
    file(classes1,t40_classes1),
    []).

fof(t42_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ~ ( ~ v4_ordinal1(A)
            & ! [B] : 
                ( v3_ordinal1(B)
               => A != k1_ordinal1(B) ) )
        & ~ ( ? [B] : 
                ( v3_ordinal1(B)
                & A = k1_ordinal1(B) )
            & v4_ordinal1(A) ) ) ) ),
    file(ordinal1,t42_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(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(t95_zfmisc_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_tarski(A),k3_tarski(B)) ) ),
    file(zfmisc_1,t95_zfmisc_1),
    []).