## TPTP Problem File: SEU704^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU704^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Conditionals
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! phi:o.! x:i.in x A -> (! y:i.in y A -> singleton
%            (dsetconstr A (^ z:i.phi & z = x | ~phi & z = y))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC206l [Bro08]

% Status   : Theorem
% Rating   : 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   15 (   0 unit;   9 type;   5 defn)
%            Number of atoms       :  119 (  18 equality;  67 variable)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :   84 (   7   ~;   5   |;  11   &;  43   @)
%                                         (   0 <=>;  18  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   11 (   9   :;   0   =)
%            Number of variables   :   27 (   0 sgn;  20   !;   1   ?;   6   ^)
%                                         (  27   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

thf(emptyset_type,type,(
emptyset: \$i )).

setadjoin: \$i > \$i > \$i )).

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

thf(singleton_type,type,(
singleton: \$i > \$o )).

thf(singleton,definition,
( singleton
= ( ^ [A: \$i] :
? [Xx: \$i] :
( ( in @ Xx @ A )
& ( A
= ( setadjoin @ Xx @ emptyset ) ) ) ) )).

thf(iffalseProp1_type,type,(
iffalseProp1: \$o )).

thf(iffalseProp1,definition,
( iffalseProp1
= ( ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ~ ( Xphi )
=> ( in @ Xy
@ ( dsetconstr @ A
@ ^ [Xz: \$i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ ( Xphi )
& ( Xz = Xy ) ) ) ) ) ) ) ) ) )).

thf(iffalseProp2_type,type,(
iffalseProp2: \$o )).

thf(iffalseProp2,definition,
( iffalseProp2
= ( ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ~ ( Xphi )
=> ( ( dsetconstr @ A
@ ^ [Xz: \$i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ ( Xphi )
& ( Xz = Xy ) ) ) )
= ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) )).

thf(iftrueProp1_type,type,(
iftrueProp1: \$o )).

thf(iftrueProp1,definition,
( iftrueProp1
= ( ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xz: \$i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ ( Xphi )
& ( Xz = Xy ) ) ) ) ) ) ) ) ) )).

thf(iftrueProp2_type,type,(
iftrueProp2: \$o )).

thf(iftrueProp2,definition,
( iftrueProp2
= ( ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( ( dsetconstr @ A
@ ^ [Xz: \$i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ ( Xphi )
& ( Xz = Xy ) ) ) )
= ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) )).

thf(ifSingleton,conjecture,
( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [Xz: \$i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ ( Xphi )
& ( Xz = Xy ) ) ) ) ) ) ) ) ) ) )).

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