%--------------------------------------------------------------------------
% File     : SET768+4 : TPTP v9.0.0. Released v2.2.0.
% Domain   : Set Theory (Equivalence relations)
% Problem  : Equality of equivalence classes 1
% Version  : [Pas99] axioms.
% English  : Two equivalence classes are equal if and only if the members
%            are equivalent.

% Refs     : [Pas99] Pastre (1999), Email to G. Sutcliffe
% Source   : [Pas99]
% Names    :

% Status   : Theorem
% Rating   : 0.67 v8.2.0, 0.78 v8.1.0, 0.69 v7.5.0, 0.75 v7.4.0, 0.73 v7.3.0, 0.72 v7.2.0, 0.69 v7.1.0, 0.57 v7.0.0, 0.63 v6.4.0, 0.62 v6.2.0, 0.52 v6.1.0, 0.57 v6.0.0, 0.65 v5.5.0, 0.78 v5.4.0, 0.79 v5.3.0, 0.78 v5.2.0, 0.75 v5.1.0, 0.81 v5.0.0, 0.83 v4.1.0, 0.87 v4.0.1, 0.91 v4.0.0, 0.92 v3.7.0, 0.90 v3.5.0, 0.89 v3.3.0, 0.86 v3.2.0, 0.91 v3.1.0, 0.89 v2.7.0, 0.83 v2.6.0, 0.86 v2.5.0, 0.88 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax   : Number of formulae    :   17 (   1 unt;   0 def)
%            Number of atoms       :   73 (   4 equ)
%            Maximal formula atoms :   13 (   4 avg)
%            Number of connectives :   59 (   3   ~;   2   |;  23   &)
%                                         (  16 <=>;  15  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   7 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 2-3 aty)
%            Number of functors    :   10 (  10 usr;   1 con; 0-3 aty)
%            Number of variables   :   61 (  57   !;   4   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%----Include set theory definitions
include('Axioms/SET006+0.ax').
%----Include equivalence relation axioms
include('Axioms/SET006+2.ax').
%--------------------------------------------------------------------------
fof(thIII04,conjecture,
    ! [E,R,A,B] :
      ( ( equivalence(R,E)
        & member(A,E)
        & member(B,E) )
     => ( equal_set(equivalence_class(A,E,R),equivalence_class(B,E,R))
      <=> apply(R,A,B) ) ) ).

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