## TPTP Problem File: SEU708^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU708^2 : 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 -> ~phi ->
%            if A phi x y = y))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC210l [Bro08]

% Status   : Theorem
% Rating   : 0.22 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v4.1.0, 0.33 v4.0.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   13 (   0 unit;   8 type;   4 defn)
%            Number of atoms       :   81 (  12 equality;  46 variable)
%            Maximal formula depth :   16 (   7 average)
%            Number of connectives :   57 (   5   ~;   3   |;   6   &;  30   @)
%                                         (   0 <=>;  13  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   11 (  11   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =)
%            Number of variables   :   21 (   0 sgn;  14   !;   0   ?;   7   ^)
%                                         (  21   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

thf(setunion_type,type,(
setunion: \$i > \$i )).

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

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

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(ifSingleton_type,type,(
ifSingleton: \$o )).

thf(ifSingleton,definition,
( ifSingleton
= ( ! [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 ) ) ) ) ) ) ) ) )).

thf(if_type,type,(
if: \$i > \$o > \$i > \$i > \$i )).

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

thf(theeq_type,type,(
theeq: \$o )).

thf(theeq,definition,
( theeq
= ( ! [X: \$i] :
( ( singleton @ X )
=> ! [Xx: \$i] :
( ( in @ Xx @ X )
=> ( ( setunion @ X )
= Xx ) ) ) ) )).

thf(iffalse,conjecture,
( iffalseProp1
=> ( ifSingleton
=> ( theeq
=> ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ~ ( Xphi )
=> ( ( if @ A @ Xphi @ Xx @ Xy )
= Xy ) ) ) ) ) ) )).

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