## TPTP Problem File: SEU611^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU611^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Operations on Sets - Symmetric Difference
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! x:i.in x (symdiff A B) -> (! phi:o.(in x A ->
%            ~(in x B) -> phi) -> (~(in x A) -> in x B -> phi) -> phi))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC113l [Bro08]

% Status   : Theorem
% Rating   : 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   12 (   0 unit;   7 type;   4 defn)
%            Number of atoms       :   69 (   4 equality;  39 variable)
%            Maximal formula depth :   15 (   6 average)
%            Number of connectives :   60 (   4   ~;   2   |;   0   &;  41   @)
%                                         (   0 <=>;  13  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   11 (  11   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   7   :;   0   =)
%            Number of variables   :   18 (   0 sgn;  13   !;   0   ?;   5   ^)
%                                         (  18   :;   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(dsetconstrER_type,type,(
dsetconstrER: \$o )).

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

thf(binunion_type,type,(
binunion: \$i > \$i > \$i )).

thf(binunionE_type,type,(
binunionE: \$o )).

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

thf(symdiff_type,type,(
symdiff: \$i > \$i > \$i )).

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

thf(symdiffE,conjecture,
( dsetconstrEL
=> ( dsetconstrER
=> ( binunionE
=> ! [A: \$i,B: \$i,Xx: \$i] :
( ( in @ Xx @ ( symdiff @ A @ B ) )
=> ! [Xphi: \$o] :
( ( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> Xphi ) )
=> ( ( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> Xphi ) )
=> Xphi ) ) ) ) ) )).

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