## TPTP Problem File: SEU726^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU726^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Typed Set Theory - Laws for Typed Sets
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
%            subset (setminus A Y) (setminus A X) -> subset X Y))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC228l [Bro08]

% Status   : Theorem
% Rating   : 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.14 v6.1.0, 0.29 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 v4.0.1, 0.33 v4.0.0, 0.00 v3.7.0
% Syntax   : Number of formulae    :   13 (   0 unit;   8 type;   4 defn)
%            Number of atoms       :   78 (   4 equality;  40 variable)
%            Maximal formula depth :   15 (   6 average)
%            Number of connectives :   67 (   2   ~;   0   |;   0   &;  48   @)
%                                         (   0 <=>;  17  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    7 (   7   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =)
%            Number of variables   :   16 (   0 sgn;  16   !;   0   ?;   0   ^)
%                                         (  16   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%          :
%------------------------------------------------------------------------------
thf(in_type,type,(
in: \$i > \$i > \$o )).

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

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

thf(subsetE_type,type,(
subsetE: \$o )).

thf(subsetE,definition,
( subsetE
= ( ! [A: \$i,B: \$i,Xx: \$i] :
( ( subset @ A @ B )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ 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(setminusER_type,type,(
setminusER: \$o )).

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

thf(subsetTI_type,type,(
subsetTI: \$o )).

thf(subsetTI,definition,
( subsetTI
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( subset @ X @ Y ) ) ) ) ) )).

thf(contrasubsetT3,conjecture,
( subsetE
=> ( setminusI
=> ( setminusER
=> ( subsetTI
=> ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
=> ( subset @ X @ Y ) ) ) ) ) ) ) )).

%------------------------------------------------------------------------------
```