%------------------------------------------------------------------------------
% File     : SET611+3 : TPTP v9.0.0. Released v2.2.0.
% Domain   : Set Theory
% Problem  : X ^ Y = the empty set iff X \ Y = X
% Version  : [Try90] axioms : Reduced > Incomplete.
% English  : The intersection of X and Y is the empty set iff the difference
%            of X and Y is 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 (84) [TS89]

% Status   : Theorem
% Rating   : 0.24 v9.0.0, 0.28 v8.2.0, 0.25 v7.5.0, 0.28 v7.4.0, 0.20 v7.3.0, 0.17 v7.2.0, 0.14 v7.1.0, 0.13 v6.4.0, 0.15 v6.3.0, 0.25 v6.2.0, 0.24 v6.1.0, 0.40 v6.0.0, 0.39 v5.5.0, 0.37 v5.4.0, 0.39 v5.3.0, 0.44 v5.2.0, 0.25 v5.1.0, 0.24 v5.0.0, 0.38 v4.1.0, 0.43 v4.0.1, 0.39 v4.0.0, 0.38 v3.7.0, 0.30 v3.5.0, 0.26 v3.4.0, 0.37 v3.3.0, 0.36 v3.2.0, 0.45 v3.1.0, 0.44 v2.7.0, 0.33 v2.6.0, 0.71 v2.5.0, 0.50 v2.4.0, 0.25 v2.3.0, 0.00 v2.2.1
% Syntax   : Number of formulae    :   11 (   3 unt;   0 def)
%            Number of atoms       :   25 (   6 equ)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   17 (   3   ~;   0   |;   3   &)
%                                         (   9 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   5 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :    3 (   3 usr;   1 con; 0-2 aty)
%            Number of variables   :   25 (  25   !;   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(3),1833060)
fof(intersection_defn,axiom,
    ! [B,C,D] :
      ( member(D,intersection(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(hidden - axiom149,1832636)
fof(empty_set_defn,axiom,
    ! [B] : ~ member(B,empty_set) ).

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

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

%---- line(hidden - axiom150,1832615)
fof(equal_member_defn,axiom,
    ! [B,C] :
      ( B = C
    <=> ! [D] :
          ( 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) ) ) ).

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

%---- line(hidden - axiom152,1832628)
fof(empty_defn,axiom,
    ! [B] :
      ( empty(B)
    <=> ! [C] : ~ member(C,B) ) ).

%---- line(boole - th(84),1834054)
fof(prove_th84,conjecture,
    ! [B,C] :
      ( intersection(B,C) = empty_set
    <=> difference(B,C) = B ) ).

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