%--------------------------------------------------------------------------
% File     : SET108+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% Domain   : Set Theory
% Problem  : 1st and 2nd make the ordered pair
% Version  : [Qua92] axioms : Reduced & Augmented > Complete.
% English  :

% Refs     : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
%          : [BL+86] Boyer et al. (1986), Set Theory in First-Order Logic:
% Source   : [Qua92]
% Names    :

% Status   : Theorem
% Rating   : 0.18 v9.0.0, 0.17 v8.1.0, 0.14 v7.5.0, 0.16 v7.4.0, 0.13 v7.3.0, 0.14 v7.1.0, 0.22 v7.0.0, 0.17 v6.4.0, 0.19 v6.3.0, 0.29 v6.2.0, 0.32 v6.1.0, 0.30 v6.0.0, 0.35 v5.5.0, 0.33 v5.4.0
% Syntax   : Number of formulae    :   44 (  16 unt;   0 def)
%            Number of atoms       :  108 (  23 equ)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives :   70 (   6   ~;   4   |;  32   &)
%                                         (  19 <=>;   9  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   5 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :    6 (   5 usr;   0 prp; 1-2 aty)
%            Number of functors    :   26 (  26 usr;   5 con; 0-3 aty)
%            Number of variables   :   91 (  82   !;   9   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
% Bugfixed : v5.4.0 - Bugfixes to SET005+0 axiom file.
%--------------------------------------------------------------------------
%----Include set theory axioms
include('Axioms/SET005+0.ax').
%--------------------------------------------------------------------------
%----OP6: Existence of 1st and 2nd.
%----All six theorems combined
fof(existence_of_first_and_second,conjecture,
    ! [X] :
    ? [U,V] :
      ( ( member(U,universal_class)
        & member(V,universal_class)
        & X = ordered_pair(U,V) )
      | ( ~ ? [Y,Z] :
              ( member(Y,universal_class)
              & member(Z,universal_class)
              & X = ordered_pair(Y,Z) )
        & U = X
        & V = X ) ) ).

%--------------------------------------------------------------------------