%------------------------------------------------------------------------------
% File     : SET580+3 : TPTP v9.0.0. Released v2.2.0.
% Domain   : Set Theory
% Problem  : x is in X sym Y iff x is in X iff x is not in Y
% Version  : [Try90] axioms : Reduced > Incomplete.
% English  : x is in the symmetric difference of X and Y iff it is not the
%            case x is in X iff x is in Y.

% 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 (23) [TS89]

% Status   : Theorem
% Rating   : 0.21 v9.0.0, 0.22 v8.2.0, 0.19 v7.5.0, 0.22 v7.4.0, 0.17 v7.3.0, 0.10 v7.2.0, 0.07 v7.1.0, 0.13 v7.0.0, 0.10 v6.4.0, 0.15 v6.3.0, 0.08 v6.2.0, 0.12 v6.1.0, 0.20 v6.0.0, 0.13 v5.5.0, 0.22 v5.4.0, 0.25 v5.3.0, 0.33 v5.2.0, 0.10 v5.1.0, 0.14 v5.0.0, 0.17 v4.1.0, 0.22 v4.0.1, 0.26 v4.0.0, 0.25 v3.5.0, 0.21 v3.4.0, 0.32 v3.3.0, 0.29 v3.2.0, 0.36 v3.1.0, 0.44 v2.7.0, 0.33 v2.6.0, 0.29 v2.5.0, 0.38 v2.4.0, 0.50 v2.3.0, 0.33 v2.2.1
% Syntax   : Number of formulae    :    7 (   3 unt;   0 def)
%            Number of atoms       :   15 (   4 equ)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   10 (   2   ~;   1   |;   1   &)
%                                         (   6 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    3 (   3 usr;   0 con; 2-2 aty)
%            Number of variables   :   18 (  18   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%------------------------------------------------------------------------------
%---- 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(boole - df(7),1833089)
fof(symmetric_difference_defn,axiom,
    ! [B,C] : symmetric_difference(B,C) = union(difference(B,C),difference(C,B)) ).

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

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

%---- line(hidden - axiom18,1832615)
fof(equal_member_defn,axiom,
    ! [B,C] :
      ( B = C
    <=> ! [D] :
          ( member(D,B)
        <=> member(D,C) ) ) ).

%---- line(boole - th(23),1833126)
fof(prove_th23,conjecture,
    ! [B,C,D] :
      ( member(B,symmetric_difference(C,D))
    <=> ~ ( member(B,C)
        <=> member(B,D) ) ) ).

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