## TPTP Problem File: SEU751^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU751^2 : TPTP v7.1.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Typed Set Theory - Laws for Typed Sets - DeMorgan Laws
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
%            (! x:i.in x A -> in x (setminus A (binintersect X Y)) ->
%            in x (binunion (setminus A X) (setminus A Y)))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC253l [Bro08]

% Status   : Theorem
% Rating   : 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, 0.29 v6.1.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   18 (   0 unit;  11 type;   6 defn)
%            Number of atoms       :  127 (   6 equality;  63 variable)
%            Maximal formula depth :   20 (   7 average)
%            Number of connectives :  114 (   6   ~;   0   |;   0   &;  83   @)
%                                         (   0 <=>;  25  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    9 (   9   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   13 (  11   :;   0   =)
%            Number of variables   :   23 (   0 sgn;  23   !;   0   ?;   0   ^)
%                                         (  23   :;   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(binunion_type,type,(
binunion: \$i > \$i > \$i )).

thf(binintersect_type,type,(
binintersect: \$i > \$i > \$i )).

thf(binintersectI_type,type,(
binintersectI: \$o )).

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

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

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(complementT_lem_type,type,(
complementT_lem: \$o )).

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

thf(complementTE1_type,type,(
complementTE1: \$o )).

thf(complementTE1,definition,
( complementTE1
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
=> ( in @ Xx @ X ) ) ) ) ) )).

thf(binunionTILcontra_type,type,(
binunionTILcontra: \$o )).

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

thf(binunionTIRcontra_type,type,(
binunionTIRcontra: \$o )).

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

thf(demorgan1a,conjecture,
( binintersectI
=> ( setminusER
=> ( complementT_lem
=> ( complementTE1
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
=> ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ) ) ) )).

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