## TPTP Problem File: SEU602^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU602^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Operations on Sets - Set Difference
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! x:i.in x (setminus A B) -> in x A)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC104l [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, 0.29 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 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :    7 (   0 unit;   4 type;   2 defn)
%            Number of atoms       :   27 (   2 equality;  14 variable)
%            Maximal formula depth :   10 (   6 average)
%            Number of connectives :   21 (   1   ~;   0   |;   0   &;  17   @)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   4   :;   0   =)
%            Number of variables   :   10 (   0 sgn;   6   !;   0   ?;   4   ^)
%                                         (  10   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

thf(dsetconstr_type,type,(
dsetconstr: \$i > ( \$i > \$o ) > \$i )).

thf(dsetconstrEL_type,type,(
dsetconstrEL: \$o )).

thf(dsetconstrEL,definition,
( dsetconstrEL
= ( ! [A: \$i,Xphi: \$i > \$o,Xx: \$i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: \$i] :
( Xphi @ Xy ) ) )
=> ( in @ Xx @ A ) ) ) )).

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

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

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

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