%------------------------------------------------------------------------------
% File     : SEU610^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Operations on Sets - Set Difference
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.setminus A B = emptyset -> subset A B)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC112l [Bro08]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v5.1.0, 0.20 v4.1.0, 0.00 v4.0.1, 0.33 v3.7.0
% Syntax   : Number of formulae    :   13 (   4 unt;   8 typ;   4 def)
%            Number of atoms       :   25 (   7 equ;   0 cnn)
%            Maximal formula atoms :    6 (   5 avg)
%            Number of connectives :   38 (   1   ~;   0   |;   0   &;  24   @)
%                                         (   1 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   8 usr;   5 con; 0-2 aty)
%            Number of variables   :   14 (   0   ^;  14   !;   0   ?;  14   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=465
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(emptyset_type,type,
    emptyset: $i ).

thf(emptysetE_type,type,
    emptysetE: $o ).

thf(emptysetE,definition,
    ( emptysetE
    = ( ! [Xx: $i] :
          ( ( in @ Xx @ emptyset )
         => ! [Xphi: $o] : Xphi ) ) ) ).

thf(in__Cong_type,type,
    in__Cong: $o ).

thf(in__Cong,definition,
    ( in__Cong
    = ( ! [A: $i,B: $i] :
          ( ( A = B )
         => ! [Xx: $i,Xy: $i] :
              ( ( Xx = Xy )
             => ( ( in @ Xx @ A )
              <=> ( in @ Xy @ B ) ) ) ) ) ) ).

thf(subset_type,type,
    subset: $i > $i > $o ).

thf(subsetI2_type,type,
    subsetI2: $o ).

thf(subsetI2,definition,
    ( subsetI2
    = ( ! [A: $i,B: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ A )
             => ( in @ Xx @ B ) )
         => ( subset @ A @ B ) ) ) ) ).

thf(setminus_type,type,
    setminus: $i > $i > $i ).

thf(setminusI_type,type,
    setminusI: $o ).

thf(setminusI,definition,
    ( setminusI
    = ( ! [A: $i,B: $i,Xx: $i] :
          ( ( in @ Xx @ A )
         => ( ~ ( in @ Xx @ B )
           => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).

thf(setminusSubset1,conjecture,
    ( emptysetE
   => ( in__Cong
     => ( subsetI2
       => ( setminusI
         => ! [A: $i,B: $i] :
              ( ( ( setminus @ A @ B )
                = emptyset )
             => ( subset @ A @ B ) ) ) ) ) ) ).

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