%--------------------------------------------------------------------------
% File     : SET595+3 : TPTP v9.0.0. Released v2.2.0.
% Domain   : Set Theory
% Problem  : If X (= Y, then Y = X U (Y \ X)
% Version  : [Try90] axioms : Reduced > Incomplete.
% English  : If X is a subset of Y, then Y is the union of X and (the
%            difference of Y and X).

% Refs     : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
%          : [Try90] Trybulec (1990), Tarski Grothendieck Set Theory
%          : [TS89]  Trybulec & Swieczkowska (1989), Boolean Properties of
% Source   : [ILF]
% Names    : BOOLE (54) [TS89]

% Status   : Theorem
% Rating   : 0.48 v9.0.0, 0.47 v8.2.0, 0.44 v8.1.0, 0.50 v7.5.0, 0.53 v7.4.0, 0.37 v7.3.0, 0.34 v7.2.0, 0.31 v7.1.0, 0.39 v7.0.0, 0.40 v6.4.0, 0.38 v6.3.0, 0.42 v6.2.0, 0.48 v6.1.0, 0.57 v5.5.0, 0.56 v5.4.0, 0.61 v5.3.0, 0.63 v5.2.0, 0.55 v5.1.0, 0.57 v5.0.0, 0.46 v4.1.0, 0.39 v4.0.1, 0.43 v4.0.0, 0.46 v3.7.0, 0.50 v3.5.0, 0.47 v3.4.0, 0.58 v3.3.0, 0.64 v3.1.0, 0.78 v2.7.0, 0.67 v2.6.0, 0.71 v2.5.0, 0.75 v2.4.0, 0.25 v2.3.0, 0.00 v2.2.1
% Syntax   : Number of formulae    :    9 (   2 unt;   0 def)
%            Number of atoms       :   22 (   5 equ)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   14 (   1   ~;   1   |;   2   &)
%                                         (   7 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 2-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 2-2 aty)
%            Number of variables   :   22 (  22   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%---- line(tarski - th(2),1832736)
fof(member_equal,axiom,
    ! [B,C] :
      ( ! [D] :
          ( member(D,B)
        <=> member(D,C) )
     => B = C ) ).

%---- line(boole - df(2),1833042)
fof(union_defn,axiom,
    ! [B,C,D] :
      ( member(D,union(B,C))
    <=> ( member(D,B)
        | member(D,C) ) ) ).

%---- line(boole - df(4),1833078)
fof(difference_defn,axiom,
    ! [B,C,D] :
      ( member(D,difference(B,C))
    <=> ( member(D,B)
        & ~ member(D,C) ) ) ).

%---- line(tarski - df(3),1832749)
fof(subset_defn,axiom,
    ! [B,C] :
      ( subset(B,C)
    <=> ! [D] :
          ( member(D,B)
         => member(D,C) ) ) ).

%---- line(boole - df(8),1833103)
fof(equal_defn,axiom,
    ! [B,C] :
      ( B = C
    <=> ( subset(B,C)
        & subset(C,B) ) ) ).

%---- property(commutativity,op(union,2,function))
fof(commutativity_of_union,axiom,
    ! [B,C] : union(B,C) = union(C,B) ).

%---- line(hidden - axiom77,1832615)
fof(equal_member_defn,axiom,
    ! [B,C] :
      ( B = C
    <=> ! [D] :
          ( member(D,B)
        <=> member(D,C) ) ) ).

%---- property(reflexivity,op(subset,2,predicate))
fof(reflexivity_of_subset,axiom,
    ! [B] : subset(B,B) ).

%---- line(boole - th(54),1833552)
fof(prove_th54,conjecture,
    ! [B,C] :
      ( subset(B,C)
     => C = union(B,difference(C,B)) ) ).

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