%--------------------------------------------------------------------------
% File     : SET587+3 : TPTP v9.0.0. Released v2.2.0.
% Domain   : Set Theory
% Problem  : X \ Y = the empty set iff X (= Y
% Version  : [Try90] axioms : Reduced > Incomplete.
% English  : The difference of X and Y is the empty set iff X is a subset of
%            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 (45) [TS89]

% Status   : Theorem
% Rating   : 0.21 v9.0.0, 0.22 v8.2.0, 0.19 v7.5.0, 0.22 v7.4.0, 0.10 v7.2.0, 0.07 v7.1.0, 0.04 v7.0.0, 0.07 v6.4.0, 0.08 v6.3.0, 0.12 v6.1.0, 0.23 v6.0.0, 0.30 v5.5.0, 0.22 v5.4.0, 0.25 v5.3.0, 0.37 v5.2.0, 0.15 v5.1.0, 0.14 v5.0.0, 0.17 v4.0.1, 0.22 v4.0.0, 0.21 v3.7.0, 0.20 v3.5.0, 0.21 v3.4.0, 0.26 v3.3.0, 0.29 v3.2.0, 0.45 v3.1.0, 0.33 v2.6.0, 0.29 v2.5.0, 0.12 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       :   21 (   4 equ)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   15 (   3   ~;   0   |;   2   &)
%                                         (   8 <=>;   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    :    2 (   2 usr;   1 con; 0-2 aty)
%            Number of variables   :   20 (  20   !;   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(4),1833078)
fof(difference_defn,axiom,
    ! [B,C,D] :
      ( member(D,difference(B,C))
    <=> ( member(D,B)
        & ~ member(D,C) ) ) ).

%---- line(hidden - axiom58,1832636)
fof(empty_set_defn,axiom,
    ! [B] : ~ member(B,empty_set) ).

%---- 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) ) ) ).

%---- line(hidden - axiom59,1832615)
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(hidden - axiom61,1832628)
fof(empty_defn,axiom,
    ! [B] :
      ( empty(B)
    <=> ! [C] : ~ member(C,B) ) ).

%---- line(boole - th(45),1833405)
fof(prove_difference_empty_set,conjecture,
    ! [B,C] :
      ( difference(B,C) = empty_set
    <=> subset(B,C) ) ).

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