## TPTP Problem File: SEU709^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU709^2 : 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 ->
%            in (if A phi x y) (setadjoin x (setadjoin y emptyset))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC211l [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 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   15 (   0 unit;   9 type;   5 defn)
%            Number of atoms       :   84 (   9 equality;  44 variable)
%            Maximal formula depth :   18 (   7 average)
%            Number of connectives :   61 (   1   ~;   0   |;   0   &;  44   @)
%                                         (   1 <=>;  15  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   11 (   9   :;   0   =)
%            Number of variables   :   20 (   0 sgn;  20   !;   0   ?;   0   ^)
%                                         (  20   :;   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 )).

= ( ! [Xx: \$i,Xy: \$i] :
( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) )).

thf(in__Cong_type,type,(
in__Cong: \$o )).

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

thf(secondinupair_type,type,(
secondinupair: \$o )).

thf(secondinupair,definition,
( secondinupair
= ( ! [Xx: \$i,Xy: \$i] :
( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) )).

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

thf(iftrue_type,type,(
iftrue: \$o )).

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

thf(iffalse_type,type,(
iffalse: \$o )).

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

thf(iftrueorfalse,conjecture,