% Mizar problem: t23_pencil_1,pencil_1,1628,46 
fof(t23_pencil_1,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] : 
              ( m1_pboole(C,A)
             => ! [D] : 
                  ( m1_pboole(D,A)
                 => ( ( r2_hidden(C,k4_card_3(B))
                      & r2_hidden(D,k4_card_3(B)) )
                   => ! [E] : 
                        ( E != k3_pencil_1(A,B)
                       => k1_funct_1(C,E) = k1_funct_1(D,E) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finset_1,cc1_realset1,cc1_yellow13,cc6_pencil_1,commutativity_k2_tarski,d1_tarski,d20_pencil_1,d21_pencil_1,d3_pboole,d4_funct_1,d4_realset1,d5_tarski,dt_k1_funct_1,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k2_tarski,dt_k3_pencil_1,dt_k4_card_3,dt_k4_tarski,dt_m1_pboole,dt_m1_subset_1,existence_m1_pboole,existence_m1_subset_1,fc1_finset_1,fc1_xboole_0,fc2_finset_1,fc2_subset_1,fc3_card_3,fc3_realset1,fc3_subset_1,fc4_pencil_1,fc5_card_3,rc1_finset_1,rc1_realset1,rc1_xboole_0,rc2_realset1,rc2_xboole_0,rc8_pencil_1,t18_card_3,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole]),
    [file(pencil_1,t23_pencil_1)]).

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

fof(cc1_finset_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    []).

fof(cc1_realset1,axiom,(
    ! [A] : 
      ( ~ v1_realset1(A)
     => ~ v1_xboole_0(A) ) ),
    file(realset1,cc1_realset1),
    []).

fof(cc1_yellow13,axiom,(
    ! [A] : 
      ( v1_realset1(A)
     => v1_finset_1(A) ) ),
    file(yellow13,cc1_yellow13),
    []).

fof(cc6_pencil_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_pboole(B,A)
         => ( ( ~ v13_pencil_1(B)
              & v16_pencil_1(B,A) )
           => v2_relat_1(B) ) ) ) ),
    file(pencil_1,cc6_pencil_1),
    []).

fof(commutativity_k2_tarski,axiom,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    []).

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

fof(d20_pencil_1,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v16_pencil_1(B,A)
      <=> ? [C] : 
            ( m1_subset_1(C,A)
            & ! [D] : 
                ( m1_subset_1(D,A)
               => ( C != D
                 => ( ~ v1_xboole_0(k1_funct_1(B,D))
                    & v1_realset1(k1_funct_1(B,D)) ) ) ) ) ) ) ),
    file(pencil_1,d20_pencil_1),
    []).

fof(d21_pencil_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] : 
              ( m1_subset_1(C,A)
             => ( C = k3_pencil_1(A,B)
              <=> ~ v1_realset1(k1_funct_1(B,C)) ) ) ) ) ),
    file(pencil_1,d21_pencil_1),
    []).

fof(d3_pboole,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( m1_pboole(B,A)
      <=> k1_relat_1(B) = A ) ) ),
    file(pboole,d3_pboole),
    []).

fof(d4_funct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( ( r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> r2_hidden(k4_tarski(B,C),A) ) )
          & ( ~ r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> C = k1_xboole_0 ) ) ) ) ),
    file(funct_1,d4_funct_1),
    []).

fof(d4_realset1,axiom,(
    ! [A] : 
      ( v1_realset1(A)
    <=> ~ ( ~ v1_xboole_0(A)
          & ! [B] : A != k1_tarski(B) ) ) ),
    file(realset1,d4_realset1),
    []).

fof(d5_tarski,axiom,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    []).

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

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_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    []).

fof(dt_k3_pencil_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v13_pencil_1(B)
        & v16_pencil_1(B,A)
        & m1_pboole(B,A) )
     => m1_subset_1(k3_pencil_1(A,B),A) ) ),
    file(pencil_1,k3_pencil_1),
    []).

fof(dt_k4_card_3,axiom,(
    $true ),
    file(card_3,k4_card_3),
    []).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    []).

fof(dt_m1_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B) ) ) ),
    file(pboole,m1_pboole),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_pboole,axiom,(
    ! [A] : 
    ? [B] : m1_pboole(B,A) ),
    file(pboole,m1_pboole),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc1_finset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A)) ) ),
    file(finset_1,fc1_finset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc2_finset_1,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(k2_tarski(A,B))
      & v1_finset_1(k2_tarski(A,B)) ) ),
    file(finset_1,fc2_finset_1),
    []).

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

fof(fc3_card_3,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => v1_fraenkel(k4_card_3(A)) ) ),
    file(card_3,fc3_card_3),
    []).

fof(fc3_realset1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A))
      & v1_realset1(k1_tarski(A)) ) ),
    file(realset1,fc3_realset1),
    []).

fof(fc3_subset_1,axiom,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ),
    file(subset_1,fc3_subset_1),
    []).

fof(fc4_pencil_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v13_pencil_1(B)
        & v16_pencil_1(B,A)
        & m1_pboole(B,A) )
     => ( ~ v1_xboole_0(k4_card_3(B))
        & v1_fraenkel(k4_card_3(B))
        & ~ v1_realset1(k4_card_3(B)) ) ) ),
    file(pencil_1,fc4_pencil_1),
    []).

fof(fc5_card_3,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A) )
     => ( ~ v1_xboole_0(k4_card_3(A))
        & v1_fraenkel(k4_card_3(A)) ) ) ),
    file(card_3,fc5_card_3),
    []).

fof(rc1_finset_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    []).

fof(rc1_realset1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_realset1(A) ) ),
    file(realset1,rc1_realset1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_realset1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & ~ v1_realset1(A) ) ),
    file(realset1,rc2_realset1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc8_pencil_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_pboole(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & ~ v13_pencil_1(B)
          & v16_pencil_1(B,A) ) ) ),
    file(pencil_1,rc8_pencil_1),
    []).

fof(t18_card_3,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r2_hidden(A,k4_card_3(B))
          <=> ( k1_relat_1(A) = k1_relat_1(B)
              & ! [C] : 
                  ( r2_hidden(C,k1_relat_1(B))
                 => r2_hidden(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) ) ),
    file(card_3,t18_card_3),
    []).

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),
    []).