## TPTP Problem File: SEU705^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU705^2 : TPTP v8.1.2. 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 ->
%            in (if A phi x y) A))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC207l [Bro08]

% Status   : Theorem
% Rating   : 0.46 v8.1.0, 0.45 v7.5.0, 0.29 v7.4.0, 0.22 v7.2.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v4.1.0, 0.33 v4.0.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   19 (   6 unt;  12 typ;   6 def)
%            Number of atoms       :   33 (  11 equ;   0 cnn)
%            Maximal formula atoms :    7 (   4 avg)
%            Number of connectives :   59 (   2   ~;   2   |;   5   &;  39   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   16 (  16   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   13 (  12 usr;   5 con; 0-4 aty)
%            Number of variables   :   23 (   8   ^  14   !;   1   ?;  23   :)
% SPC      : TH0_THM_EQU_NAR

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

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

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

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

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

thf(subset_type,type,
subset: \$i > \$i > \$o ).

thf(subsetE_type,type,
subsetE: \$o ).

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

thf(sepSubset_type,type,
sepSubset: \$o ).

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

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

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

thf(theprop_type,type,
theprop: \$o ).

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

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(ifp,conjecture,
( subsetE
=> ( sepSubset
=> ( theprop
=> ( ifSingleton
=> ! [A: \$i,Xphi: \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ) ) ).

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