%------------------------------------------------------------------------------
% File     : SEU714^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Typed Set Theory - Types of Set Operators
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! X:i.in X (powerset A) -> in (setminus A X) (powerset A))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC216l [Bro08]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.00 v3.7.0
% Syntax   : Number of formulae    :    8 (   2 unt;   5 typ;   2 def)
%            Number of atoms       :   13 (   2 equ;   0 cnn)
%            Maximal formula atoms :    4 (   4 avg)
%            Number of connectives :   27 (   0   ~;   0   |;   0   &;  21   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   4 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   5 usr;   2 con; 0-2 aty)
%            Number of variables   :    8 (   0   ^;   8   !;   0   ?;   8   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=272
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(powerset_type,type,
    powerset: $i > $i ).

thf(powersetI_type,type,
    powersetI: $o ).

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

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

thf(setminusEL_type,type,
    setminusEL: $o ).

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

thf(complementT_lem,conjecture,
    ( powersetI
   => ( setminusEL
     => ! [A: $i,X: $i] :
          ( ( in @ X @ ( powerset @ A ) )
         => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).

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