SET007 Axioms: SET007+69.ax


%------------------------------------------------------------------------------
% File     : SET007+69 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Increasing and Continuous Ordinal Sequences
% 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    : ordinal4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (   3 unit)
%            Number of atoms       :  490 (  50 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  462 (  37 ~  ;   8  |; 238  &)
%                                         (   3 <=>; 176 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-2 arity)
%            Number of functors    :   31 (   4 constant; 0-3 arity)
%            Number of variables   :  160 (   0 singleton; 153 !;   7 ?)
%            Maximal term depth    :    4 (   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_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( v1_ordinal1(k1_ordinal2(A))
        & v2_ordinal1(k1_ordinal2(A))
        & v3_ordinal1(k1_ordinal2(A)) ) ) )).

fof(fc2_ordinal4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => ( v1_relat_1(k1_ordinal4(A,B))
        & v1_funct_1(k1_ordinal4(A,B))
        & v5_ordinal1(k1_ordinal4(A,B))
        & v1_ordinal2(k1_ordinal4(A,B)) ) ) )).

fof(rc1_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] :
          ( m1_ordinal4(B,A)
          & v1_ordinal1(B)
          & v2_ordinal1(B)
          & v3_ordinal1(B)
          & ~ v1_xboole_0(B) ) ) )).

fof(t1_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k1_relat_1(A) = k1_ordinal1(B)
           => ( r1_ordinal2(k1_ordinal2(A),A)
              & k12_ordinal2(A) = k1_ordinal2(A) ) ) ) ) )).

fof(d1_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C) )
             => ( C = k1_ordinal4(A,B)
              <=> ( k1_relat_1(C) = k14_ordinal2(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(A))
                       => k1_funct_1(C,D) = k1_funct_1(A,D) ) )
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(C,k14_ordinal2(k1_relat_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) )).

fof(t2_ordinal4,axiom,(
    $true )).

fof(t3_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal2(C,A)
               => r1_ordinal2(C,k1_ordinal4(B,A)) ) ) ) ) )).

fof(t4_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal2(B,A)
               => r1_ordinal2(k14_ordinal2(C,B),k1_ordinal3(C,A)) ) ) ) ) )).

fof(t5_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal2(B,A)
               => r1_ordinal2(k15_ordinal2(B,C),k4_ordinal3(C,A)) ) ) ) ) )).

fof(t6_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( k1_relat_1(A) = k1_relat_1(B)
                      & r1_ordinal2(C,A)
                      & r1_ordinal2(D,B) )
                   => ( ( ? [E] :
                            ( v3_ordinal1(E)
                            & r2_hidden(E,k1_relat_1(A))
                            & ~ r1_ordinal1(k1_funct_1(A,E),k1_funct_1(B,E)) )
                        & ? [E] :
                            ( v3_ordinal1(E)
                            & r2_hidden(E,k1_relat_1(A))
                            & ~ r2_hidden(k1_funct_1(A,E),k1_funct_1(B,E)) ) )
                      | r1_ordinal1(C,D) ) ) ) ) ) ) )).

fof(t7_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_ordinal2(C) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D) )
                 => ( ( k1_relat_1(C) = k1_relat_1(A)
                      & k1_relat_1(A) = k1_relat_1(D)
                      & r1_ordinal2(B,C)
                      & r1_ordinal2(B,D)
                      & ! [E] :
                          ( v3_ordinal1(E)
                         => ( r2_hidden(E,k1_relat_1(A))
                           => ( r1_ordinal1(k1_funct_1(C,E),k1_funct_1(A,E))
                              & r1_ordinal1(k1_funct_1(A,E),k1_funct_1(D,E)) ) ) ) )
                   => r1_ordinal2(B,A) ) ) ) ) ) )).

fof(t8_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( ( v4_ordinal1(k1_relat_1(A))
          & v2_ordinal2(A) )
       => ( k1_relat_1(A) = k1_xboole_0
          | ( r1_ordinal2(k8_ordinal2(A),A)
            & k12_ordinal2(A) = k8_ordinal2(A) ) ) ) ) )).

fof(t9_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( ( v2_ordinal2(A)
                  & r1_ordinal1(B,C)
                  & r2_hidden(C,k1_relat_1(A)) )
               => r1_ordinal1(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) )).

fof(t10_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( ( v2_ordinal2(A)
              & r2_hidden(B,k1_relat_1(A)) )
           => r1_ordinal1(B,k1_funct_1(A,B)) ) ) ) )).

fof(t11_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v2_ordinal2(A)
           => v3_ordinal1(k10_relat_1(A,B)) ) ) ) )).

fof(t12_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( v2_ordinal2(A)
           => ( v1_relat_1(k5_relat_1(A,B))
              & v1_funct_1(k5_relat_1(A,B))
              & v5_ordinal1(k5_relat_1(A,B))
              & v1_ordinal2(k5_relat_1(A,B)) ) ) ) ) )).

fof(t13_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ~ ( v2_ordinal2(A)
              & v2_ordinal2(B)
              & ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ~ ( C = k5_relat_1(B,A)
                      & v2_ordinal2(C) ) ) ) ) ) )).

fof(t14_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_ordinal2(C) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D) )
                 => ( ( v2_ordinal2(C)
                      & r1_ordinal2(B,D)
                      & k7_ordinal2(k2_relat_1(C)) = k1_relat_1(D)
                      & A = k5_relat_1(C,D) )
                   => r1_ordinal2(B,A) ) ) ) ) ) )).

fof(t15_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v2_ordinal2(A)
           => v2_ordinal2(k2_ordinal1(A,B)) ) ) ) )).

fof(t16_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( ( v2_ordinal2(A)
          & v4_ordinal1(k1_relat_1(A)) )
       => v4_ordinal1(k8_ordinal2(A)) ) ) )).

fof(t17_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_ordinal2(C) )
             => ( ( v2_ordinal2(A)
                  & v3_ordinal2(A)
                  & v3_ordinal2(B)
                  & C = k5_relat_1(A,B) )
               => v3_ordinal2(C) ) ) ) ) )).

fof(t18_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( ! [C] :
                ( v3_ordinal1(C)
               => ( r2_hidden(C,k1_relat_1(A))
                 => k1_funct_1(A,C) = k14_ordinal2(B,C) ) )
           => v2_ordinal2(A) ) ) ) )).

fof(t19_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( ! [C] :
                ( v3_ordinal1(C)
               => ( r2_hidden(C,k1_relat_1(A))
                 => k1_funct_1(A,C) = k15_ordinal2(C,B) ) )
           => ( B = k1_xboole_0
              | v2_ordinal2(A) ) ) ) ) )).

fof(t20_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( A != k1_xboole_0
       => k16_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ) )).

fof(t21_ordinal4,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) ) ) )
                   => r1_ordinal2(k16_ordinal2(B,A),C) ) ) ) ) ) ) )).

fof(t22_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ~ ( A != k1_xboole_0
              & k16_ordinal2(A,B) = k1_xboole_0 ) ) ) )).

fof(t23_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(k4_ordinal2,A)
           => r2_hidden(k16_ordinal2(A,B),k16_ordinal2(A,k1_ordinal1(B))) ) ) ) )).

fof(t24_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( ( r2_hidden(k4_ordinal2,A)
                  & r2_hidden(B,C) )
               => r2_hidden(k16_ordinal2(A,B),k16_ordinal2(A,C)) ) ) ) ) )).

fof(t25_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( ( r2_hidden(k4_ordinal2,B)
              & ! [C] :
                  ( v3_ordinal1(C)
                 => ( r2_hidden(C,k1_relat_1(A))
                   => k1_funct_1(A,C) = k16_ordinal2(B,C) ) ) )
           => v2_ordinal2(A) ) ) ) )).

fof(t26_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( ( r2_hidden(k4_ordinal2,A)
              & v4_ordinal1(B) )
           => ( B = k1_xboole_0
              | ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = B
                      & ! [D] :
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,B)
                           => k1_funct_1(C,D) = k16_ordinal2(A,D) ) ) )
                   => k16_ordinal2(A,B) = k8_ordinal2(C) ) ) ) ) ) ) )).

fof(t27_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(B,C)
               => ( A = k1_xboole_0
                  | r1_ordinal1(k16_ordinal2(A,B),k16_ordinal2(A,C)) ) ) ) ) ) )).

fof(t28_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k16_ordinal2(A,C),k16_ordinal2(B,C)) ) ) ) ) )).

fof(t29_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(k4_ordinal2,A)
           => ( B = k1_xboole_0
              | r2_hidden(k4_ordinal2,k16_ordinal2(A,B)) ) ) ) ) )).

fof(t30_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k16_ordinal2(A,k14_ordinal2(B,C)) = k15_ordinal2(k16_ordinal2(A,C),k16_ordinal2(A,B)) ) ) ) )).

fof(t31_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k16_ordinal2(k16_ordinal2(A,B),C) = k16_ordinal2(A,k15_ordinal2(C,B)) ) ) ) )).

fof(t32_ordinal4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(k4_ordinal2,A)
           => r1_ordinal1(B,k16_ordinal2(A,B)) ) ) ) )).

fof(d2_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( m1_ordinal4(B,A)
          <=> r2_hidden(B,A) ) ) ) )).

fof(d3_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( m2_ordinal4(B,A)
          <=> ( k1_relat_1(B) = k2_ordinal2(A)
              & r1_tarski(k2_relat_1(B),k2_ordinal2(A)) ) ) ) ) )).

fof(d4_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => k2_ordinal4(A) = k1_xboole_0 ) )).

fof(d5_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => k3_ordinal4(A) = k4_ordinal2 ) )).

fof(t33_ordinal4,axiom,(
    $true )).

fof(t34_ordinal4,axiom,(
    $true )).

fof(t35_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( k2_ordinal4(A) = k1_xboole_0
        & k3_ordinal4(A) = k4_ordinal2 ) ) )).

fof(t36_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal4(B,A)
         => ! [C] :
              ( m2_ordinal4(C,A)
             => r2_hidden(B,k1_relat_1(C)) ) ) ) )).

fof(t37_ordinal4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ( ( r2_hidden(k1_relat_1(A),B)
              & r1_tarski(k2_relat_1(A),B) )
           => r2_hidden(k8_ordinal2(A),B) ) ) ) )).

fof(t38_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m2_ordinal4(B,A)
         => ~ ( v2_ordinal2(B)
              & v3_ordinal2(B)
              & r2_hidden(k5_ordinal2,A)
              & ! [C] :
                  ( v3_ordinal1(C)
                 => ~ ( r2_hidden(C,k1_relat_1(B))
                      & k1_funct_1(B,C) = C ) ) ) ) ) )).

fof(s1_ordinal4,axiom,
    ( ( ! [A] :
          ( v3_ordinal1(A)
         => ! [B] :
              ( v3_ordinal1(B)
             => ( r2_hidden(A,B)
               => r2_hidden(f1_s1_ordinal4(A),f1_s1_ordinal4(B)) ) ) )
      & ! [A] :
          ( ( v1_relat_1(A)
            & v1_funct_1(A)
            & v5_ordinal1(A)
            & v1_ordinal2(A) )
         => ! [B] :
              ( v3_ordinal1(B)
             => ( v4_ordinal1(B)
               => ( B = k1_xboole_0
                  | ! [C] :
                      ( ( v1_relat_1(C)
                        & v1_funct_1(C)
                        & v5_ordinal1(C)
                        & v1_ordinal2(C) )
                     => ( ( k1_relat_1(A) = B
                          & ! [D] :
                              ( v3_ordinal1(D)
                             => ( r2_hidden(D,B)
                               => k1_funct_1(C,D) = f1_s1_ordinal4(D) ) ) )
                       => r1_ordinal2(f1_s1_ordinal4(B),C) ) ) ) ) ) ) )
   => ? [A] :
        ( v3_ordinal1(A)
        & f1_s1_ordinal4(A) = A ) )).

fof(s2_ordinal4,axiom,(
    ? [A] :
      ( m2_ordinal4(A,f1_s2_ordinal4)
      & ! [B] :
          ( m1_ordinal4(B,f1_s2_ordinal4)
         => k1_funct_1(A,B) = f2_s2_ordinal4(B) ) ) )).

fof(dt_m1_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal4(B,A)
         => v3_ordinal1(B) ) ) )).

fof(existence_m1_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] : m1_ordinal4(B,A) ) )).

fof(dt_m2_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m2_ordinal4(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) ) ) ) )).

fof(existence_m2_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] : m2_ordinal4(B,A) ) )).

fof(dt_k1_ordinal4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B) )
     => ( v1_relat_1(k1_ordinal4(A,B))
        & v1_funct_1(k1_ordinal4(A,B))
        & v5_ordinal1(k1_ordinal4(A,B)) ) ) )).

fof(dt_k2_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => m1_ordinal4(k2_ordinal4(A),A) ) )).

fof(dt_k3_ordinal4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( ~ v1_xboole_0(k3_ordinal4(A))
        & m1_ordinal4(k3_ordinal4(A),A) ) ) )).

fof(dt_k4_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m2_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => m1_ordinal4(k4_ordinal4(A,B,C),A) ) )).

fof(redefinition_k4_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m2_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => k4_ordinal4(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k5_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m2_ordinal4(B,A)
        & m2_ordinal4(C,A) )
     => m2_ordinal4(k5_ordinal4(A,B,C),A) ) )).

fof(redefinition_k5_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m2_ordinal4(B,A)
        & m2_ordinal4(C,A) )
     => k5_ordinal4(A,B,C) = k5_relat_1(B,C) ) )).

fof(dt_k6_ordinal4,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A) )
     => ( ~ v1_xboole_0(k6_ordinal4(A,B))
        & m1_ordinal4(k6_ordinal4(A,B),A) ) ) )).

fof(redefinition_k6_ordinal4,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A) )
     => k6_ordinal4(A,B) = k1_ordinal1(B) ) )).

fof(dt_k7_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => m1_ordinal4(k7_ordinal4(A,B,C),A) ) )).

fof(redefinition_k7_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => k7_ordinal4(A,B,C) = k14_ordinal2(B,C) ) )).

fof(dt_k8_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => m1_ordinal4(k8_ordinal4(A,B,C),A) ) )).

fof(redefinition_k8_ordinal4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => k8_ordinal4(A,B,C) = k15_ordinal2(B,C) ) )).
%------------------------------------------------------------------------------