SET007 Axioms: SET007+27.ax


%------------------------------------------------------------------------------
% File     : SET007+27 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Ordinal Arithmetics
% 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    : ordinal3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  101 (  17 unit)
%            Number of atoms       :  515 ( 110 equality)
%            Maximal formula depth :   16 (   7 average)
%            Number of connectives :  448 (  34 ~  ;  16  |; 135  &)
%                                         (   9 <=>; 254 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-2 arity)
%            Number of functors    :   22 (   2 constant; 0-2 arity)
%            Number of variables   :  219 (   0 singleton; 218 !;   1 ?)
%            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_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( v1_ordinal1(k2_xboole_0(A,B))
        & v2_ordinal1(k2_xboole_0(A,B))
        & v3_ordinal1(k2_xboole_0(A,B)) ) ) )).

fof(fc2_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( v1_ordinal1(k3_xboole_0(A,B))
        & v2_ordinal1(k3_xboole_0(A,B))
        & v3_ordinal1(k3_xboole_0(A,B)) ) ) )).

fof(t1_ordinal3,axiom,(
    ! [A] : r1_tarski(A,k1_ordinal1(A)) )).

fof(t2_ordinal3,axiom,(
    ! [A,B] :
      ( r1_tarski(k1_ordinal1(A),B)
     => r1_tarski(A,B) ) )).

fof(t3_ordinal3,axiom,(
    $true )).

fof(t4_ordinal3,axiom,(
    $true )).

fof(t5_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k1_ordinal1(A),k1_ordinal1(B)) ) ) ) )).

fof(t6_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => v3_ordinal1(k3_tarski(B)) ) ) )).

fof(t7_ordinal3,axiom,(
    ! [A] : v3_ordinal1(k3_tarski(k2_ordinal2(A))) )).

fof(t8_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => k2_ordinal2(B) = B ) ) )).

fof(t9_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k2_ordinal2(k1_tarski(A)) = k1_tarski(A) ) )).

fof(t10_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( A != k1_xboole_0
       => r2_hidden(k1_xboole_0,A) ) ) )).

fof(t11_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k6_ordinal2(A) = k1_xboole_0 ) )).

fof(t12_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k6_ordinal2(k1_tarski(A)) = A ) )).

fof(t13_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => v3_ordinal1(k1_setfam_1(B)) ) ) )).

fof(t14_ordinal3,axiom,(
    $true )).

fof(t15_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k2_xboole_0(A,B) = A
            | k2_xboole_0(A,B) = B ) ) ) )).

fof(t16_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k3_xboole_0(A,B) = A
            | k3_xboole_0(A,B) = B ) ) ) )).

fof(t17_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( r2_hidden(A,k4_ordinal2)
       => A = k1_xboole_0 ) ) )).

fof(t18_ordinal3,axiom,(
    k4_ordinal2 = k1_tarski(k1_xboole_0) )).

fof(t19_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ~ ( r1_ordinal1(A,k4_ordinal2)
          & A != k1_xboole_0
          & A != k4_ordinal2 ) ) )).

fof(t20_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ( r2_hidden(C,D)
                   => ( ( ~ r1_ordinal1(A,B)
                        & ~ r2_hidden(A,B) )
                      | r2_hidden(k14_ordinal2(A,C),k14_ordinal2(B,D)) ) ) ) ) ) ) )).

fof(t21_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( r1_ordinal1(A,B)
                      & r1_ordinal1(C,D) )
                   => r1_ordinal1(k14_ordinal2(A,C),k14_ordinal2(B,D)) ) ) ) ) ) )).

fof(t22_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ( r2_hidden(A,B)
                   => ( ( ~ ( r1_ordinal1(C,D)
                            & D != k1_xboole_0 )
                        & ~ r2_hidden(C,D) )
                      | r2_hidden(k15_ordinal2(A,C),k15_ordinal2(B,D)) ) ) ) ) ) ) )).

fof(t23_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( r1_ordinal1(A,B)
                      & r1_ordinal1(C,D) )
                   => r1_ordinal1(k15_ordinal2(A,C),k15_ordinal2(B,D)) ) ) ) ) ) )).

fof(t24_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( k14_ordinal2(A,B) = k14_ordinal2(A,C)
               => B = C ) ) ) ) )).

fof(t25_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(k14_ordinal2(A,B),k14_ordinal2(A,C))
               => r2_hidden(B,C) ) ) ) ) )).

fof(t26_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(k14_ordinal2(A,B),k14_ordinal2(A,C))
               => r1_ordinal1(B,C) ) ) ) ) )).

fof(t27_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,k14_ordinal2(A,B))
            & r1_ordinal1(B,k14_ordinal2(A,B)) ) ) ) )).

fof(t28_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(A,B)
               => ( r2_hidden(A,k14_ordinal2(B,C))
                  & r2_hidden(A,k14_ordinal2(C,B)) ) ) ) ) ) )).

fof(t29_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k14_ordinal2(A,B) = k1_xboole_0
           => ( A = k1_xboole_0
              & B = k1_xboole_0 ) ) ) ) )).

fof(t30_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ~ ( r1_ordinal1(A,B)
              & ! [C] :
                  ( v3_ordinal1(C)
                 => B != k14_ordinal2(A,C) ) ) ) ) )).

fof(t31_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ~ ( r2_hidden(A,B)
              & ! [C] :
                  ( v3_ordinal1(C)
                 => ~ ( B = k14_ordinal2(A,C)
                      & C != k1_xboole_0 ) ) ) ) ) )).

fof(t32_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | v4_ordinal1(k14_ordinal2(B,A)) ) ) ) ) )).

fof(t33_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k14_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(A,k14_ordinal2(B,C)) ) ) ) )).

fof(t34_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ~ ( k15_ordinal2(A,B) = k1_xboole_0
              & A != k1_xboole_0
              & B != k1_xboole_0 ) ) ) )).

fof(t35_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(A,B)
               => ( C = k1_xboole_0
                  | ( r2_hidden(A,k15_ordinal2(B,C))
                    & r2_hidden(A,k15_ordinal2(C,B)) ) ) ) ) ) ) )).

fof(t36_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( k15_ordinal2(A,B) = k15_ordinal2(C,B)
               => ( B = k1_xboole_0
                  | A = C ) ) ) ) ) )).

fof(t37_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(k15_ordinal2(A,B),k15_ordinal2(C,B))
               => r2_hidden(A,C) ) ) ) ) )).

fof(t38_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(k15_ordinal2(A,B),k15_ordinal2(C,B))
               => ( B = k1_xboole_0
                  | r1_ordinal1(A,C) ) ) ) ) ) )).

fof(t39_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( A != k1_xboole_0
           => ( r1_ordinal1(B,k15_ordinal2(B,A))
              & r1_ordinal1(B,k15_ordinal2(A,B)) ) ) ) ) )).

fof(t40_ordinal3,axiom,(
    $true )).

fof(t41_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k15_ordinal2(A,B) = k4_ordinal2
           => ( A = k4_ordinal2
              & B = k4_ordinal2 ) ) ) ) )).

fof(t42_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ~ ( r2_hidden(A,k14_ordinal2(B,C))
                  & ~ r2_hidden(A,B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ~ ( r2_hidden(D,C)
                          & A = k14_ordinal2(B,D) ) ) ) ) ) ) )).

fof(d1_ordinal3,axiom,(
    $true )).

fof(d2_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(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) )
             => ( C = k1_ordinal3(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(C,D) = k14_ordinal2(A,k1_funct_1(B,D)) ) ) ) ) ) ) ) )).

fof(d3_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(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) )
             => ( C = k2_ordinal3(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(C,D) = k14_ordinal2(k1_funct_1(B,D),A) ) ) ) ) ) ) ) )).

fof(d4_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(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) )
             => ( C = k3_ordinal3(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(C,D) = k15_ordinal2(A,k1_funct_1(B,D)) ) ) ) ) ) ) ) )).

fof(d5_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(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) )
             => ( C = k4_ordinal3(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(C,D) = k15_ordinal2(k1_funct_1(B,D),A) ) ) ) ) ) ) ) )).

fof(t43_ordinal3,axiom,(
    $true )).

fof(t44_ordinal3,axiom,(
    $true )).

fof(t45_ordinal3,axiom,(
    $true )).

fof(t46_ordinal3,axiom,(
    $true )).

fof(t47_ordinal3,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)
             => ( ( k1_relat_1(A) = k1_relat_1(B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ! [E] :
                          ( v3_ordinal1(E)
                         => ( ( r2_hidden(D,k1_relat_1(A))
                              & E = k1_funct_1(A,D) )
                           => k1_funct_1(B,D) = k14_ordinal2(C,E) ) ) ) )
               => ( k1_xboole_0 = k1_relat_1(A)
                  | k8_ordinal2(B) = k14_ordinal2(C,k8_ordinal2(A)) ) ) ) ) ) )).

fof(t48_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => v4_ordinal1(k15_ordinal2(A,B)) ) ) ) )).

fof(t49_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ~ ( r2_hidden(A,k15_ordinal2(B,C))
                  & v4_ordinal1(B)
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ~ ( r2_hidden(D,B)
                          & r2_hidden(A,k15_ordinal2(D,C)) ) ) ) ) ) ) )).

fof(t50_ordinal3,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)
             => ( ( k1_relat_1(A) = k1_relat_1(B)
                  & v4_ordinal1(k8_ordinal2(A))
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ! [E] :
                          ( v3_ordinal1(E)
                         => ( ( r2_hidden(D,k1_relat_1(A))
                              & E = k1_funct_1(A,D) )
                           => k1_funct_1(B,D) = k15_ordinal2(E,C) ) ) ) )
               => ( k1_xboole_0 = k1_relat_1(A)
                  | C = k1_xboole_0
                  | k8_ordinal2(B) = k15_ordinal2(k8_ordinal2(A),C) ) ) ) ) ) )).

fof(t51_ordinal3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k1_xboole_0 != k1_relat_1(A)
           => k8_ordinal2(k1_ordinal3(B,A)) = k14_ordinal2(B,k8_ordinal2(A)) ) ) ) )).

fof(t52_ordinal3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v4_ordinal1(k8_ordinal2(A))
           => ( k1_xboole_0 = k1_relat_1(A)
              | B = k1_xboole_0
              | k8_ordinal2(k4_ordinal3(B,A)) = k15_ordinal2(k8_ordinal2(A),B) ) ) ) ) )).

fof(t53_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( A != k1_xboole_0
           => k3_tarski(k14_ordinal2(B,A)) = k14_ordinal2(B,k3_tarski(A)) ) ) ) )).

fof(t54_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k15_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(k15_ordinal2(A,C),k15_ordinal2(B,C)) ) ) ) )).

fof(t55_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ~ ( A != k1_xboole_0
              & ! [C] :
                  ( v3_ordinal1(C)
                 => ! [D] :
                      ( v3_ordinal1(D)
                     => ~ ( B = k14_ordinal2(k15_ordinal2(C,A),D)
                          & r2_hidden(D,A) ) ) ) ) ) ) )).

fof(t56_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ! [E] :
                      ( v3_ordinal1(E)
                     => ( ( k14_ordinal2(k15_ordinal2(B,A),C) = k14_ordinal2(k15_ordinal2(D,A),E)
                          & r2_hidden(C,A)
                          & r2_hidden(E,A) )
                       => ( B = D
                          & C = E ) ) ) ) ) ) ) )).

fof(t57_ordinal3,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) = k15_ordinal2(D,A) ) ) )
                   => k15_ordinal2(B,A) = k8_ordinal2(C) ) ) ) ) ) ) )).

fof(t58_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k15_ordinal2(k15_ordinal2(A,B),C) = k15_ordinal2(A,k15_ordinal2(B,C)) ) ) ) )).

fof(d6_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( ( r1_ordinal1(B,A)
                 => ( C = k5_ordinal3(A,B)
                  <=> A = k14_ordinal2(B,C) ) )
                & ( ~ r1_ordinal1(B,A)
                 => ( C = k5_ordinal3(A,B)
                  <=> C = k1_xboole_0 ) ) ) ) ) ) )).

fof(d7_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( ( B != k1_xboole_0
                 => ( C = k6_ordinal3(A,B)
                  <=> ? [D] :
                        ( v3_ordinal1(D)
                        & A = k14_ordinal2(k15_ordinal2(C,B),D)
                        & r2_hidden(D,B) ) ) )
                & ( B = k1_xboole_0
                 => ( C = k6_ordinal3(A,B)
                  <=> C = k1_xboole_0 ) ) ) ) ) ) )).

fof(d8_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k7_ordinal3(A,B) = k5_ordinal3(A,k15_ordinal2(k6_ordinal3(A,B),B)) ) ) )).

fof(t59_ordinal3,axiom,(
    $true )).

fof(t60_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
           => B = k14_ordinal2(A,k5_ordinal3(B,A)) ) ) ) )).

fof(t61_ordinal3,axiom,(
    $true )).

fof(t62_ordinal3,axiom,(
    $true )).

fof(t63_ordinal3,axiom,(
    $true )).

fof(t64_ordinal3,axiom,(
    $true )).

fof(t65_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k5_ordinal3(k14_ordinal2(A,B),A) = B ) ) )).

fof(t66_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(A,B)
               => ( ( ~ r1_ordinal1(C,A)
                    & ~ r2_hidden(C,A) )
                  | r2_hidden(k5_ordinal3(A,C),k5_ordinal3(B,C)) ) ) ) ) ) )).

fof(t67_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k5_ordinal3(A,A) = k1_xboole_0 ) )).

fof(t68_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
           => ( k5_ordinal3(B,A) != k1_xboole_0
              & r2_hidden(k1_xboole_0,k5_ordinal3(B,A)) ) ) ) ) )).

fof(t69_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( k5_ordinal3(A,k1_xboole_0) = A
        & k5_ordinal3(k1_xboole_0,A) = k1_xboole_0 ) ) )).

fof(t70_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k5_ordinal3(A,k14_ordinal2(B,C)) = k5_ordinal3(k5_ordinal3(A,B),C) ) ) ) )).

fof(t71_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k5_ordinal3(C,B),k5_ordinal3(C,A)) ) ) ) ) )).

fof(t72_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k5_ordinal3(A,C),k5_ordinal3(B,C)) ) ) ) ) )).

fof(t73_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(B,k14_ordinal2(C,A))
               => ( A = k1_xboole_0
                  | r2_hidden(k5_ordinal3(B,C),A) ) ) ) ) ) )).

fof(t74_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(k14_ordinal2(A,B),C)
               => r2_hidden(B,k5_ordinal3(C,A)) ) ) ) ) )).

fof(t75_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => r1_ordinal1(A,k14_ordinal2(B,k5_ordinal3(A,B))) ) ) )).

fof(t76_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => k5_ordinal3(k15_ordinal2(A,B),k15_ordinal2(C,B)) = k15_ordinal2(k5_ordinal3(A,C),B) ) ) ) )).

fof(t77_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => r1_ordinal1(k15_ordinal2(k6_ordinal3(A,B),B),A) ) ) )).

fof(t78_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => A = k14_ordinal2(k15_ordinal2(k6_ordinal3(A,B),B),k7_ordinal3(A,B)) ) ) )).

fof(t79_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( A = k14_ordinal2(k15_ordinal2(B,C),D)
                      & r2_hidden(D,C) )
                   => ( B = k6_ordinal3(A,C)
                      & D = k7_ordinal3(A,C) ) ) ) ) ) ) )).

fof(t80_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r2_hidden(A,k15_ordinal2(B,C))
               => ( r2_hidden(k6_ordinal3(A,C),B)
                  & r2_hidden(k7_ordinal3(A,C),C) ) ) ) ) ) )).

fof(t81_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( A != k1_xboole_0
           => k6_ordinal3(k15_ordinal2(B,A),A) = B ) ) ) )).

fof(t82_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k7_ordinal3(k15_ordinal2(A,B),B) = k1_xboole_0 ) ) )).

fof(t83_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( k6_ordinal3(k1_xboole_0,A) = k1_xboole_0
        & k7_ordinal3(k1_xboole_0,A) = k1_xboole_0
        & k7_ordinal3(A,k1_xboole_0) = A ) ) )).

fof(t84_ordinal3,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( k6_ordinal3(A,k4_ordinal2) = A
        & k7_ordinal3(A,k4_ordinal2) = k1_xboole_0 ) ) )).

fof(dt_k1_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => ( v1_relat_1(k1_ordinal3(A,B))
        & v1_funct_1(k1_ordinal3(A,B))
        & v5_ordinal1(k1_ordinal3(A,B))
        & v1_ordinal2(k1_ordinal3(A,B)) ) ) )).

fof(dt_k2_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => ( v1_relat_1(k2_ordinal3(A,B))
        & v1_funct_1(k2_ordinal3(A,B))
        & v5_ordinal1(k2_ordinal3(A,B))
        & v1_ordinal2(k2_ordinal3(A,B)) ) ) )).

fof(dt_k3_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => ( v1_relat_1(k3_ordinal3(A,B))
        & v1_funct_1(k3_ordinal3(A,B))
        & v5_ordinal1(k3_ordinal3(A,B))
        & v1_ordinal2(k3_ordinal3(A,B)) ) ) )).

fof(dt_k4_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => ( v1_relat_1(k4_ordinal3(A,B))
        & v1_funct_1(k4_ordinal3(A,B))
        & v5_ordinal1(k4_ordinal3(A,B))
        & v1_ordinal2(k4_ordinal3(A,B)) ) ) )).

fof(dt_k5_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k5_ordinal3(A,B)) ) )).

fof(dt_k6_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k6_ordinal3(A,B)) ) )).

fof(dt_k7_ordinal3,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k7_ordinal3(A,B)) ) )).
%------------------------------------------------------------------------------