SET007 Axioms: SET007+26.ax


%------------------------------------------------------------------------------
% File     : SET007+26 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Sequences of Ordinal Numbers
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : ordinal2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  125 (  29 unit)
%            Number of atoms       :  834 ( 226 equality)
%            Maximal formula depth :   23 (   7 average)
%            Number of connectives :  722 (  13 ~  ;  30  |; 344  &)
%                                         (  27 <=>; 308 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-2 arity)
%            Number of functors    :   95 (  33 constant; 0-2 arity)
%            Number of variables   :  274 (   0 singleton; 240 !;  34 ?)
%            Maximal term depth    :    6 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_ordinal2,axiom,
    ( v1_ordinal1(k5_ordinal2)
    & v2_ordinal1(k5_ordinal2)
    & v3_ordinal1(k5_ordinal2)
    & ~ v1_xboole_0(k5_ordinal2) )).

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

fof(rc2_ordinal2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & v1_ordinal2(A) ) )).

fof(cc1_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( m1_ordinal1(B,A)
         => v1_ordinal2(B) ) ) )).

fof(fc2_ordinal2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A)
        & v3_ordinal1(B) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B))
        & v1_ordinal2(k7_relat_1(A,B)) ) ) )).

fof(fc3_ordinal2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A)
        & v3_ordinal1(B) )
     => ( v1_ordinal1(k1_funct_1(A,B))
        & v2_ordinal1(k1_funct_1(A,B))
        & v3_ordinal1(k1_funct_1(A,B)) ) ) )).

fof(fc4_ordinal2,axiom,(
    ! [A,B] :
      ( v3_ordinal1(B)
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B))
        & v1_ordinal2(k2_funcop_1(A,B)) ) ) )).

fof(t1_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
          <=> r1_ordinal1(k1_ordinal1(A),k1_ordinal1(B)) ) ) ) )).

fof(t2_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k3_tarski(k1_ordinal1(A)) = A ) )).

fof(t3_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => r1_tarski(k1_ordinal1(A),k1_zfmisc_1(A)) ) )).

fof(t4_ordinal2,axiom,(
    v4_ordinal1(k1_xboole_0) )).

fof(t5_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => r1_ordinal1(k3_tarski(A),A) ) )).

fof(d1_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => k1_ordinal2(A) = k1_funct_1(A,k3_tarski(k1_relat_1(A))) ) )).

fof(t6_ordinal2,axiom,(
    $true )).

fof(t7_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B) )
         => ( k1_relat_1(B) = k1_ordinal1(A)
           => k1_ordinal2(B) = k1_funct_1(B,A) ) ) ) )).

fof(d2_ordinal2,axiom,(
    ! [A,B] :
      ( B = k2_ordinal2(A)
    <=> ! [C] :
          ( r2_hidden(C,B)
        <=> ( r2_hidden(C,A)
            & v3_ordinal1(C) ) ) ) )).

fof(d3_ordinal2,axiom,(
    ! [A,B] :
      ( B = k3_ordinal2(A)
    <=> ! [C] :
          ( r2_hidden(C,B)
        <=> ( r2_hidden(C,A)
            & ? [D] :
                ( v3_ordinal1(D)
                & C = D
                & v4_ordinal1(D) ) ) ) ) )).

fof(t8_ordinal2,axiom,(
    $true )).

fof(t9_ordinal2,axiom,(
    ! [A] : r1_tarski(k2_ordinal2(A),A) )).

fof(t10_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k2_ordinal2(A) = A ) )).

fof(t11_ordinal2,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_tarski(k2_ordinal2(A),k2_ordinal2(B)) ) )).

fof(t12_ordinal2,axiom,(
    $true )).

fof(t13_ordinal2,axiom,(
    ! [A] : r1_tarski(k3_ordinal2(A),A) )).

fof(t14_ordinal2,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_tarski(k3_ordinal2(A),k3_ordinal2(B)) ) )).

fof(t15_ordinal2,axiom,(
    ! [A] : r1_tarski(k3_ordinal2(A),k2_ordinal2(A)) )).

fof(t16_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ? [B] :
          ( v3_ordinal1(B)
          & r2_hidden(A,B)
          & v4_ordinal1(B) ) ) )).

fof(t17_ordinal2,axiom,(
    ! [A] :
      ( ! [B] :
          ( r2_hidden(B,A)
         => v3_ordinal1(B) )
     => v3_ordinal1(k1_setfam_1(A)) ) )).

fof(d4_ordinal2,axiom,(
    k4_ordinal2 = k1_ordinal1(k1_xboole_0) )).

fof(d5_ordinal2,axiom,(
    ! [A] :
      ( A = k5_ordinal2
    <=> ( r2_hidden(k1_xboole_0,A)
        & v4_ordinal1(A)
        & v3_ordinal1(A)
        & ! [B] :
            ( v3_ordinal1(B)
           => ( ( r2_hidden(k1_xboole_0,B)
                & v4_ordinal1(B) )
             => r1_tarski(A,B) ) ) ) ) )).

fof(d6_ordinal2,axiom,(
    ! [A] : k6_ordinal2(A) = k1_setfam_1(k2_ordinal2(A)) )).

fof(d7_ordinal2,axiom,(
    ! [A,B] :
      ( v3_ordinal1(B)
     => ( B = k7_ordinal2(A)
      <=> ( r1_tarski(k2_ordinal2(A),B)
          & ! [C] :
              ( v3_ordinal1(C)
             => ( r1_tarski(k2_ordinal2(A),C)
               => r1_ordinal1(B,C) ) ) ) ) ) )).

fof(t18_ordinal2,axiom,(
    $true )).

fof(t19_ordinal2,axiom,
    ( r2_hidden(k1_xboole_0,k5_ordinal2)
    & v4_ordinal1(k5_ordinal2)
    & ! [A] :
        ( v3_ordinal1(A)
       => ( ( r2_hidden(k1_xboole_0,A)
            & v4_ordinal1(A) )
         => r1_ordinal1(k5_ordinal2,A) ) ) )).

fof(t20_ordinal2,axiom,(
    $true )).

fof(t21_ordinal2,axiom,(
    $true )).

fof(t22_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r2_hidden(A,B)
         => r1_ordinal1(k6_ordinal2(B),A) ) ) )).

fof(t23_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(C,B)
               => r1_ordinal1(A,C) ) )
         => ( k2_ordinal2(B) = k1_xboole_0
            | r1_ordinal1(A,k6_ordinal2(B)) ) ) ) )).

fof(t24_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B,C] :
          ( ( r2_hidden(A,B)
            & r1_tarski(B,C) )
         => r1_ordinal1(k6_ordinal2(C),k6_ordinal2(B)) ) ) )).

fof(t25_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r2_hidden(A,B)
         => r2_hidden(k6_ordinal2(B),B) ) ) )).

fof(t26_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k7_ordinal2(A) = A ) )).

fof(t27_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r2_hidden(A,B)
         => r2_hidden(A,k7_ordinal2(B)) ) ) )).

fof(t28_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(C,B)
               => r2_hidden(C,A) ) )
         => r1_ordinal1(k7_ordinal2(B),A) ) ) )).

fof(t29_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ~ ( r2_hidden(A,k7_ordinal2(B))
            & ! [C] :
                ( v3_ordinal1(C)
               => ~ ( r2_hidden(C,B)
                    & r1_ordinal1(A,C) ) ) ) ) )).

fof(t30_ordinal2,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_ordinal1(k7_ordinal2(A),k7_ordinal2(B)) ) )).

fof(t31_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k7_ordinal2(k1_tarski(A)) = k1_ordinal1(A) ) )).

fof(t32_ordinal2,axiom,(
    ! [A] : r1_ordinal1(k6_ordinal2(A),k7_ordinal2(A)) )).

fof(d8_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_ordinal2(A)
      <=> ? [B] :
            ( v3_ordinal1(B)
            & r1_tarski(k2_relat_1(A),B) ) ) ) )).

fof(t33_ordinal2,axiom,(
    $true )).

fof(t34_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( r2_hidden(A,k1_relat_1(B))
           => v3_ordinal1(k1_funct_1(B,A)) ) ) ) )).

fof(d9_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A)) ) )).

fof(d10_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => k9_ordinal2(A) = k6_ordinal2(k2_relat_1(A)) ) )).

fof(t35_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ( k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A))
        & k9_ordinal2(A) = k6_ordinal2(k2_relat_1(A)) ) ) )).

fof(d11_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( B = k10_ordinal2(A)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_ordinal2(C)
                & B = k9_ordinal2(C)
                & k1_relat_1(C) = k1_relat_1(A)
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( r2_hidden(D,k1_relat_1(A))
                     => k1_funct_1(C,D) = k7_ordinal2(k2_relat_1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),D)))) ) ) ) ) ) ) )).

fof(d12_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( B = k11_ordinal2(A)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_ordinal2(C)
                & B = k8_ordinal2(C)
                & k1_relat_1(C) = k1_relat_1(A)
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( r2_hidden(D,k1_relat_1(A))
                     => k1_funct_1(C,D) = k6_ordinal2(k2_relat_1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),D)))) ) ) ) ) ) ) )).

fof(d13_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( ( A = k1_xboole_0
             => ( r1_ordinal2(A,B)
              <=> ? [C] :
                    ( v3_ordinal1(C)
                    & r2_hidden(C,k1_relat_1(B))
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( ( r1_ordinal1(C,D)
                            & r2_hidden(D,k1_relat_1(B)) )
                         => k1_funct_1(B,D) = k1_xboole_0 ) ) ) ) )
            & ( A != k1_xboole_0
             => ( r1_ordinal2(A,B)
              <=> ! [C] :
                    ( v3_ordinal1(C)
                   => ! [D] :
                        ( v3_ordinal1(D)
                       => ~ ( r2_hidden(C,A)
                            & r2_hidden(A,D)
                            & ! [E] :
                                ( v3_ordinal1(E)
                               => ~ ( r2_hidden(E,k1_relat_1(B))
                                    & ! [F] :
                                        ( v3_ordinal1(F)
                                       => ( ( r1_ordinal1(E,F)
                                            & r2_hidden(F,k1_relat_1(B)) )
                                         => ( r2_hidden(C,k1_funct_1(B,F))
                                            & r2_hidden(k1_funct_1(B,F),D) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d14_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( ? [B] :
            ( v3_ordinal1(B)
            & r1_ordinal2(B,A) )
       => ! [B] :
            ( v3_ordinal1(B)
           => ( B = k12_ordinal2(A)
            <=> r1_ordinal2(B,A) ) ) ) ) )).

fof(d15_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => k13_ordinal2(A,B) = k12_ordinal2(k2_ordinal1(B,A)) ) ) )).

fof(d16_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( v2_ordinal2(A)
      <=> ! [B] :
            ( v3_ordinal1(B)
           => ! [C] :
                ( v3_ordinal1(C)
               => ( ( r2_hidden(B,C)
                    & r2_hidden(C,k1_relat_1(A)) )
                 => r2_hidden(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) ) )).

fof(d17_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( v3_ordinal2(A)
      <=> ! [B] :
            ( v3_ordinal1(B)
           => ! [C] :
                ( v3_ordinal1(C)
               => ( ( r2_hidden(B,k1_relat_1(A))
                    & v4_ordinal1(B)
                    & C = k1_funct_1(A,B) )
                 => ( B = k1_xboole_0
                    | r1_ordinal2(C,k2_ordinal1(A,B)) ) ) ) ) ) ) )).

fof(d18_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( C = k14_ordinal2(A,B)
              <=> ? [D] :
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D)
                    & C = k1_ordinal2(D)
                    & k1_relat_1(D) = k1_ordinal1(B)
                    & k1_funct_1(D,k1_xboole_0) = A
                    & ! [E] :
                        ( v3_ordinal1(E)
                       => ( r2_hidden(k1_ordinal1(E),k1_ordinal1(B))
                         => k1_funct_1(D,k1_ordinal1(E)) = k1_ordinal1(k1_funct_1(D,E)) ) )
                    & ! [E] :
                        ( v3_ordinal1(E)
                       => ( ( r2_hidden(E,k1_ordinal1(B))
                            & v4_ordinal1(E) )
                         => ( E = k1_xboole_0
                            | k1_funct_1(D,E) = k8_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) )).

fof(d19_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( C = k15_ordinal2(A,B)
              <=> ? [D] :
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D)
                    & C = k1_ordinal2(D)
                    & k1_relat_1(D) = k1_ordinal1(A)
                    & k1_funct_1(D,k1_xboole_0) = k1_xboole_0
                    & ! [E] :
                        ( v3_ordinal1(E)
                       => ( r2_hidden(k1_ordinal1(E),k1_ordinal1(A))
                         => k1_funct_1(D,k1_ordinal1(E)) = k14_ordinal2(k1_funct_1(D,E),B) ) )
                    & ! [E] :
                        ( v3_ordinal1(E)
                       => ( ( r2_hidden(E,k1_ordinal1(A))
                            & v4_ordinal1(E) )
                         => ( E = k1_xboole_0
                            | k1_funct_1(D,E) = k3_tarski(k8_ordinal2(k2_ordinal1(D,E))) ) ) ) ) ) ) ) ) )).

fof(d20_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( C = k16_ordinal2(A,B)
              <=> ? [D] :
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D)
                    & C = k1_ordinal2(D)
                    & k1_relat_1(D) = k1_ordinal1(B)
                    & k1_funct_1(D,k1_xboole_0) = k4_ordinal2
                    & ! [E] :
                        ( v3_ordinal1(E)
                       => ( r2_hidden(k1_ordinal1(E),k1_ordinal1(B))
                         => k1_funct_1(D,k1_ordinal1(E)) = k15_ordinal2(A,k1_funct_1(D,E)) ) )
                    & ! [E] :
                        ( v3_ordinal1(E)
                       => ( ( r2_hidden(E,k1_ordinal1(B))
                            & v4_ordinal1(E) )
                         => ( E = k1_xboole_0
                            | k1_funct_1(D,E) = k12_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) )).

fof(t36_ordinal2,axiom,(
    $true )).

fof(t37_ordinal2,axiom,(
    $true )).

fof(t38_ordinal2,axiom,(
    $true )).

fof(t39_ordinal2,axiom,(
    $true )).

fof(t40_ordinal2,axiom,(
    $true )).

fof(t41_ordinal2,axiom,(
    $true )).

fof(t42_ordinal2,axiom,(
    $true )).

fof(t43_ordinal2,axiom,(
    $true )).

fof(t44_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k14_ordinal2(A,k1_xboole_0) = A ) )).

fof(t45_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k14_ordinal2(A,k1_ordinal1(B)) = k1_ordinal1(k14_ordinal2(A,B)) ) ) )).

fof(t46_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = A
                      & ! [D] :
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,A)
                           => k1_funct_1(C,D) = k14_ordinal2(B,D) ) ) )
                   => k14_ordinal2(B,A) = k8_ordinal2(C) ) ) ) ) ) ) )).

fof(t47_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k14_ordinal2(k1_xboole_0,A) = A ) )).

fof(t48_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k14_ordinal2(A,k4_ordinal2) = k1_ordinal1(A) ) )).

fof(t49_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(A,B)
               => r2_hidden(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) )).

fof(t50_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) )).

fof(t51_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k14_ordinal2(A,C),k14_ordinal2(B,C)) ) ) ) ) )).

fof(t52_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k15_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) )).

fof(t53_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k15_ordinal2(k1_ordinal1(A),B) = k14_ordinal2(k15_ordinal2(A,B),B) ) ) )).

fof(t54_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = A
                      & ! [D] :
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,A)
                           => k1_funct_1(C,D) = k15_ordinal2(D,B) ) ) )
                   => k15_ordinal2(A,B) = k3_tarski(k8_ordinal2(C)) ) ) ) ) ) ) )).

fof(t55_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k15_ordinal2(A,k1_xboole_0) = k1_xboole_0 ) )).

fof(t56_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( k15_ordinal2(k4_ordinal2,A) = A
        & k15_ordinal2(A,k4_ordinal2) = A ) ) )).

fof(t57_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(B,C)
               => ( A = k1_xboole_0
                  | r2_hidden(k15_ordinal2(B,A),k15_ordinal2(C,A)) ) ) ) ) ) )).

fof(t58_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k15_ordinal2(A,C),k15_ordinal2(B,C)) ) ) ) ) )).

fof(t59_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k15_ordinal2(C,A),k15_ordinal2(C,B)) ) ) ) ) )).

fof(t60_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k16_ordinal2(A,k1_xboole_0) = k4_ordinal2 ) )).

fof(t61_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k16_ordinal2(A,k1_ordinal1(B)) = k15_ordinal2(A,k16_ordinal2(A,B)) ) ) )).

fof(t62_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = A
                      & ! [D] :
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,A)
                           => k1_funct_1(C,D) = k16_ordinal2(B,D) ) ) )
                   => k16_ordinal2(B,A) = k12_ordinal2(C) ) ) ) ) ) ) )).

fof(t63_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( k16_ordinal2(A,k4_ordinal2) = A
        & k16_ordinal2(k4_ordinal2,A) = k4_ordinal2 ) ) )).

fof(d21_ordinal2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
    <=> r2_hidden(A,k5_ordinal2) ) )).

fof(t64_ordinal2,axiom,(
    $true )).

fof(t65_ordinal2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ? [B] :
          ( v3_ordinal1(B)
          & ? [C] :
              ( v3_ordinal1(C)
              & v4_ordinal1(B)
              & v4_ordinal2(C)
              & A = k14_ordinal2(B,C) ) ) ) )).

fof(s1_ordinal2,axiom,
    ( ( p1_s1_ordinal2(k1_xboole_0)
      & ! [A] :
          ( v3_ordinal1(A)
         => ( p1_s1_ordinal2(A)
           => p1_s1_ordinal2(k1_ordinal1(A)) ) )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( ( v4_ordinal1(A)
              & ! [B] :
                  ( v3_ordinal1(B)
                 => ( r2_hidden(B,A)
                   => p1_s1_ordinal2(B) ) ) )
           => ( A = k1_xboole_0
              | p1_s1_ordinal2(A) ) ) ) )
   => ! [A] :
        ( v3_ordinal1(A)
       => p1_s1_ordinal2(A) ) )).

fof(s2_ordinal2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & k1_relat_1(A) = f1_s2_ordinal2
      & ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(B,f1_s2_ordinal2)
           => k1_funct_1(A,B) = f2_s2_ordinal2(B) ) ) ) )).

fof(s3_ordinal2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & v1_ordinal2(A)
      & k1_relat_1(A) = f1_s3_ordinal2
      & ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(B,f1_s3_ordinal2)
           => k1_funct_1(A,B) = f2_s3_ordinal2(B) ) ) ) )).

fof(s4_ordinal2,axiom,
    ( ( k1_relat_1(f5_s4_ordinal2) = f1_s4_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s4_ordinal2)
       => k1_funct_1(f5_s4_ordinal2,k1_xboole_0) = f2_s4_ordinal2 )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f1_s4_ordinal2)
           => k1_funct_1(f5_s4_ordinal2,k1_ordinal1(A)) = f3_s4_ordinal2(A,k1_funct_1(f5_s4_ordinal2,A)) ) )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f1_s4_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f5_s4_ordinal2,A) = f4_s4_ordinal2(A,k2_ordinal1(f5_s4_ordinal2,A)) ) ) )
      & k1_relat_1(f6_s4_ordinal2) = f1_s4_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s4_ordinal2)
       => k1_funct_1(f6_s4_ordinal2,k1_xboole_0) = f2_s4_ordinal2 )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f1_s4_ordinal2)
           => k1_funct_1(f6_s4_ordinal2,k1_ordinal1(A)) = f3_s4_ordinal2(A,k1_funct_1(f6_s4_ordinal2,A)) ) )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f1_s4_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f6_s4_ordinal2,A) = f4_s4_ordinal2(A,k2_ordinal1(f6_s4_ordinal2,A)) ) ) ) )
   => f5_s4_ordinal2 = f6_s4_ordinal2 )).

fof(s5_ordinal2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & k1_relat_1(A) = f1_s5_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s5_ordinal2)
       => k1_funct_1(A,k1_xboole_0) = f2_s5_ordinal2 )
      & ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(k1_ordinal1(B),f1_s5_ordinal2)
           => k1_funct_1(A,k1_ordinal1(B)) = f3_s5_ordinal2(B,k1_funct_1(A,B)) ) )
      & ! [B] :
          ( v3_ordinal1(B)
         => ( ( r2_hidden(B,f1_s5_ordinal2)
              & v4_ordinal1(B) )
           => ( B = k1_xboole_0
              | k1_funct_1(A,B) = f4_s5_ordinal2(B,k2_ordinal1(A,B)) ) ) ) ) )).

fof(s6_ordinal2,axiom,
    ( ( ! [A] :
          ( v3_ordinal1(A)
         => ! [B] :
              ( B = f2_s6_ordinal2(A)
            <=> ? [C] :
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & B = k1_ordinal2(C)
                  & k1_relat_1(C) = k1_ordinal1(A)
                  & k1_funct_1(C,k1_xboole_0) = f4_s6_ordinal2
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f5_s6_ordinal2(D,k1_funct_1(C,D)) ) )
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( ( r2_hidden(D,k1_ordinal1(A))
                          & v4_ordinal1(D) )
                       => ( D = k1_xboole_0
                          | k1_funct_1(C,D) = f6_s6_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
      & k1_relat_1(f1_s6_ordinal2) = f3_s6_ordinal2
      & ( r2_hidden(k1_xboole_0,f3_s6_ordinal2)
       => k1_funct_1(f1_s6_ordinal2,k1_xboole_0) = f4_s6_ordinal2 )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f3_s6_ordinal2)
           => k1_funct_1(f1_s6_ordinal2,k1_ordinal1(A)) = f5_s6_ordinal2(A,k1_funct_1(f1_s6_ordinal2,A)) ) )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f3_s6_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f1_s6_ordinal2,A) = f6_s6_ordinal2(A,k2_ordinal1(f1_s6_ordinal2,A)) ) ) ) )
   => ! [A] :
        ( v3_ordinal1(A)
       => ( r2_hidden(A,k1_relat_1(f1_s6_ordinal2))
         => k1_funct_1(f1_s6_ordinal2,A) = f2_s6_ordinal2(A) ) ) )).

fof(s7_ordinal2,axiom,
    ( ? [A,B] :
        ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & A = k1_ordinal2(B)
        & k1_relat_1(B) = k1_ordinal1(f1_s7_ordinal2)
        & k1_funct_1(B,k1_xboole_0) = f2_s7_ordinal2
        & ! [C] :
            ( v3_ordinal1(C)
           => ( r2_hidden(k1_ordinal1(C),k1_ordinal1(f1_s7_ordinal2))
             => k1_funct_1(B,k1_ordinal1(C)) = f3_s7_ordinal2(C,k1_funct_1(B,C)) ) )
        & ! [C] :
            ( v3_ordinal1(C)
           => ( ( r2_hidden(C,k1_ordinal1(f1_s7_ordinal2))
                & v4_ordinal1(C) )
             => ( C = k1_xboole_0
                | k1_funct_1(B,C) = f4_s7_ordinal2(C,k2_ordinal1(B,C)) ) ) ) )
    & ! [A,B] :
        ~ ( ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v5_ordinal1(C)
              & A = k1_ordinal2(C)
              & k1_relat_1(C) = k1_ordinal1(f1_s7_ordinal2)
              & k1_funct_1(C,k1_xboole_0) = f2_s7_ordinal2
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s7_ordinal2))
                   => k1_funct_1(C,k1_ordinal1(D)) = f3_s7_ordinal2(D,k1_funct_1(C,D)) ) )
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( r2_hidden(D,k1_ordinal1(f1_s7_ordinal2))
                      & v4_ordinal1(D) )
                   => ( D = k1_xboole_0
                      | k1_funct_1(C,D) = f4_s7_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
          & ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v5_ordinal1(C)
              & B = k1_ordinal2(C)
              & k1_relat_1(C) = k1_ordinal1(f1_s7_ordinal2)
              & k1_funct_1(C,k1_xboole_0) = f2_s7_ordinal2
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s7_ordinal2))
                   => k1_funct_1(C,k1_ordinal1(D)) = f3_s7_ordinal2(D,k1_funct_1(C,D)) ) )
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( r2_hidden(D,k1_ordinal1(f1_s7_ordinal2))
                      & v4_ordinal1(D) )
                   => ( D = k1_xboole_0
                      | k1_funct_1(C,D) = f4_s7_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
          & A != B ) )).

fof(s8_ordinal2,axiom,
    ( ! [A] :
        ( v3_ordinal1(A)
       => ! [B] :
            ( B = f1_s8_ordinal2(A)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & B = k1_ordinal2(C)
                & k1_relat_1(C) = k1_ordinal1(A)
                & k1_funct_1(C,k1_xboole_0) = f2_s8_ordinal2
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                     => k1_funct_1(C,k1_ordinal1(D)) = f3_s8_ordinal2(D,k1_funct_1(C,D)) ) )
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( ( r2_hidden(D,k1_ordinal1(A))
                        & v4_ordinal1(D) )
                     => ( D = k1_xboole_0
                        | k1_funct_1(C,D) = f4_s8_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
   => f1_s8_ordinal2(k1_xboole_0) = f2_s8_ordinal2 )).

fof(s9_ordinal2,axiom,
    ( ! [A] :
        ( v3_ordinal1(A)
       => ! [B] :
            ( B = f4_s9_ordinal2(A)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & B = k1_ordinal2(C)
                & k1_relat_1(C) = k1_ordinal1(A)
                & k1_funct_1(C,k1_xboole_0) = f1_s9_ordinal2
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                     => k1_funct_1(C,k1_ordinal1(D)) = f2_s9_ordinal2(D,k1_funct_1(C,D)) ) )
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( ( r2_hidden(D,k1_ordinal1(A))
                        & v4_ordinal1(D) )
                     => ( D = k1_xboole_0
                        | k1_funct_1(C,D) = f3_s9_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
   => ! [A] :
        ( v3_ordinal1(A)
       => f4_s9_ordinal2(k1_ordinal1(A)) = f2_s9_ordinal2(A,f4_s9_ordinal2(A)) ) )).

fof(s10_ordinal2,axiom,
    ( ( ! [A] :
          ( v3_ordinal1(A)
         => ! [B] :
              ( B = f3_s10_ordinal2(A)
            <=> ? [C] :
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & B = k1_ordinal2(C)
                  & k1_relat_1(C) = k1_ordinal1(A)
                  & k1_funct_1(C,k1_xboole_0) = f4_s10_ordinal2
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f5_s10_ordinal2(D,k1_funct_1(C,D)) ) )
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( ( r2_hidden(D,k1_ordinal1(A))
                          & v4_ordinal1(D) )
                       => ( D = k1_xboole_0
                          | k1_funct_1(C,D) = f6_s10_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
      & f2_s10_ordinal2 != k1_xboole_0
      & v4_ordinal1(f2_s10_ordinal2)
      & k1_relat_1(f1_s10_ordinal2) = f2_s10_ordinal2
      & ! [A] :
          ( v3_ordinal1(A)
         => ( r2_hidden(A,f2_s10_ordinal2)
           => k1_funct_1(f1_s10_ordinal2,A) = f3_s10_ordinal2(A) ) ) )
   => f3_s10_ordinal2(f2_s10_ordinal2) = f6_s10_ordinal2(f2_s10_ordinal2,f1_s10_ordinal2) )).

fof(s11_ordinal2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & v1_ordinal2(A)
      & k1_relat_1(A) = f1_s11_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s11_ordinal2)
       => k1_funct_1(A,k1_xboole_0) = f2_s11_ordinal2 )
      & ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(k1_ordinal1(B),f1_s11_ordinal2)
           => k1_funct_1(A,k1_ordinal1(B)) = f3_s11_ordinal2(B,k1_funct_1(A,B)) ) )
      & ! [B] :
          ( v3_ordinal1(B)
         => ( ( r2_hidden(B,f1_s11_ordinal2)
              & v4_ordinal1(B) )
           => ( B = k1_xboole_0
              | k1_funct_1(A,B) = f4_s11_ordinal2(B,k2_ordinal1(A,B)) ) ) ) ) )).

fof(s12_ordinal2,axiom,
    ( ( ! [A] :
          ( v3_ordinal1(A)
         => ! [B] :
              ( v3_ordinal1(B)
             => ( B = f2_s12_ordinal2(A)
              <=> ? [C] :
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & B = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(A)
                    & k1_funct_1(C,k1_xboole_0) = f4_s12_ordinal2
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                         => k1_funct_1(C,k1_ordinal1(D)) = f5_s12_ordinal2(D,k1_funct_1(C,D)) ) )
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( ( r2_hidden(D,k1_ordinal1(A))
                            & v4_ordinal1(D) )
                         => ( D = k1_xboole_0
                            | k1_funct_1(C,D) = f6_s12_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
      & k1_relat_1(f1_s12_ordinal2) = f3_s12_ordinal2
      & ( r2_hidden(k1_xboole_0,f3_s12_ordinal2)
       => k1_funct_1(f1_s12_ordinal2,k1_xboole_0) = f4_s12_ordinal2 )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f3_s12_ordinal2)
           => k1_funct_1(f1_s12_ordinal2,k1_ordinal1(A)) = f5_s12_ordinal2(A,k1_funct_1(f1_s12_ordinal2,A)) ) )
      & ! [A] :
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f3_s12_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f1_s12_ordinal2,A) = f6_s12_ordinal2(A,k2_ordinal1(f1_s12_ordinal2,A)) ) ) ) )
   => ! [A] :
        ( v3_ordinal1(A)
       => ( r2_hidden(A,k1_relat_1(f1_s12_ordinal2))
         => k1_funct_1(f1_s12_ordinal2,A) = f2_s12_ordinal2(A) ) ) )).

fof(s13_ordinal2,axiom,
    ( ? [A] :
        ( v3_ordinal1(A)
        & ? [B] :
            ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B)
            & A = k1_ordinal2(B)
            & k1_relat_1(B) = k1_ordinal1(f1_s13_ordinal2)
            & k1_funct_1(B,k1_xboole_0) = f2_s13_ordinal2
            & ! [C] :
                ( v3_ordinal1(C)
               => ( r2_hidden(k1_ordinal1(C),k1_ordinal1(f1_s13_ordinal2))
                 => k1_funct_1(B,k1_ordinal1(C)) = f3_s13_ordinal2(C,k1_funct_1(B,C)) ) )
            & ! [C] :
                ( v3_ordinal1(C)
               => ( ( r2_hidden(C,k1_ordinal1(f1_s13_ordinal2))
                    & v4_ordinal1(C) )
                 => ( C = k1_xboole_0
                    | k1_funct_1(B,C) = f4_s13_ordinal2(C,k2_ordinal1(B,C)) ) ) ) ) )
    & ! [A] :
        ( v3_ordinal1(A)
       => ! [B] :
            ( v3_ordinal1(B)
           => ~ ( ? [C] :
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & A = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(f1_s13_ordinal2)
                    & k1_funct_1(C,k1_xboole_0) = f2_s13_ordinal2
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s13_ordinal2))
                         => k1_funct_1(C,k1_ordinal1(D)) = f3_s13_ordinal2(D,k1_funct_1(C,D)) ) )
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( ( r2_hidden(D,k1_ordinal1(f1_s13_ordinal2))
                            & v4_ordinal1(D) )
                         => ( D = k1_xboole_0
                            | k1_funct_1(C,D) = f4_s13_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
                & ? [C] :
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & B = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(f1_s13_ordinal2)
                    & k1_funct_1(C,k1_xboole_0) = f2_s13_ordinal2
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s13_ordinal2))
                         => k1_funct_1(C,k1_ordinal1(D)) = f3_s13_ordinal2(D,k1_funct_1(C,D)) ) )
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( ( r2_hidden(D,k1_ordinal1(f1_s13_ordinal2))
                            & v4_ordinal1(D) )
                         => ( D = k1_xboole_0
                            | k1_funct_1(C,D) = f4_s13_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
                & A != B ) ) ) )).

fof(s14_ordinal2,axiom,
    ( ! [A] :
        ( v3_ordinal1(A)
       => ! [B] :
            ( v3_ordinal1(B)
           => ( B = f1_s14_ordinal2(A)
            <=> ? [C] :
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & v1_ordinal2(C)
                  & B = k1_ordinal2(C)
                  & k1_relat_1(C) = k1_ordinal1(A)
                  & k1_funct_1(C,k1_xboole_0) = f2_s14_ordinal2
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f3_s14_ordinal2(D,k1_funct_1(C,D)) ) )
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( ( r2_hidden(D,k1_ordinal1(A))
                          & v4_ordinal1(D) )
                       => ( D = k1_xboole_0
                          | k1_funct_1(C,D) = f4_s14_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
   => f1_s14_ordinal2(k1_xboole_0) = f2_s14_ordinal2 )).

fof(s15_ordinal2,axiom,
    ( ! [A] :
        ( v3_ordinal1(A)
       => ! [B] :
            ( v3_ordinal1(B)
           => ( B = f4_s15_ordinal2(A)
            <=> ? [C] :
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & v1_ordinal2(C)
                  & B = k1_ordinal2(C)
                  & k1_relat_1(C) = k1_ordinal1(A)
                  & k1_funct_1(C,k1_xboole_0) = f1_s15_ordinal2
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f2_s15_ordinal2(D,k1_funct_1(C,D)) ) )
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( ( r2_hidden(D,k1_ordinal1(A))
                          & v4_ordinal1(D) )
                       => ( D = k1_xboole_0
                          | k1_funct_1(C,D) = f3_s15_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
   => ! [A] :
        ( v3_ordinal1(A)
       => f4_s15_ordinal2(k1_ordinal1(A)) = f2_s15_ordinal2(A,f4_s15_ordinal2(A)) ) )).

fof(s16_ordinal2,axiom,
    ( ( ! [A] :
          ( v3_ordinal1(A)
         => ! [B] :
              ( v3_ordinal1(B)
             => ( B = f3_s16_ordinal2(A)
              <=> ? [C] :
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & B = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(A)
                    & k1_funct_1(C,k1_xboole_0) = f4_s16_ordinal2
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                         => k1_funct_1(C,k1_ordinal1(D)) = f5_s16_ordinal2(D,k1_funct_1(C,D)) ) )
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ( ( r2_hidden(D,k1_ordinal1(A))
                            & v4_ordinal1(D) )
                         => ( D = k1_xboole_0
                            | k1_funct_1(C,D) = f6_s16_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
      & f2_s16_ordinal2 != k1_xboole_0
      & v4_ordinal1(f2_s16_ordinal2)
      & k1_relat_1(f1_s16_ordinal2) = f2_s16_ordinal2
      & ! [A] :
          ( v3_ordinal1(A)
         => ( r2_hidden(A,f2_s16_ordinal2)
           => k1_funct_1(f1_s16_ordinal2,A) = f3_s16_ordinal2(A) ) ) )
   => f3_s16_ordinal2(f2_s16_ordinal2) = f6_s16_ordinal2(f2_s16_ordinal2,f1_s16_ordinal2) )).

fof(dt_k1_ordinal2,axiom,(
    $true )).

fof(dt_k2_ordinal2,axiom,(
    $true )).

fof(dt_k3_ordinal2,axiom,(
    $true )).

fof(dt_k4_ordinal2,axiom,
    ( v3_ordinal1(k4_ordinal2)
    & ~ v1_xboole_0(k4_ordinal2) )).

fof(dt_k5_ordinal2,axiom,(
    $true )).

fof(dt_k6_ordinal2,axiom,(
    ! [A] : v3_ordinal1(k6_ordinal2(A)) )).

fof(dt_k7_ordinal2,axiom,(
    ! [A] : v3_ordinal1(k7_ordinal2(A)) )).

fof(dt_k8_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => v3_ordinal1(k8_ordinal2(A)) ) )).

fof(dt_k9_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => v3_ordinal1(k9_ordinal2(A)) ) )).

fof(dt_k10_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => v3_ordinal1(k10_ordinal2(A)) ) )).

fof(dt_k11_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => v3_ordinal1(k11_ordinal2(A)) ) )).

fof(dt_k12_ordinal2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => v3_ordinal1(k12_ordinal2(A)) ) )).

fof(dt_k13_ordinal2,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => v3_ordinal1(k13_ordinal2(A,B)) ) )).

fof(dt_k14_ordinal2,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k14_ordinal2(A,B)) ) )).

fof(dt_k15_ordinal2,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k15_ordinal2(A,B)) ) )).

fof(dt_k16_ordinal2,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k16_ordinal2(A,B)) ) )).
%------------------------------------------------------------------------------