## TPTP Problem File: SEU811^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU811^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Omega and Peano
% Version  : Especial > Reduced > Especial.
% English  : (! x:i.in x omega -> (! y:i.in y omega -> omegaS x = omegaS y ->
%            x = y))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC313l [Bro08]

% Status   : Theorem
% Rating   : 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   14 (   0 unit;   8 type;   5 defn)
%            Number of atoms       :   68 (  10 equality;  33 variable)
%            Maximal formula depth :   12 (   5 average)
%            Number of connectives :   43 (   1   ~;   0   |;   0   &;  26   @)
%                                         (   1 <=>;  15  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =)
%            Number of variables   :   15 (   0 sgn;  14   !;   0   ?;   1   ^)
%                                         (  15   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

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

thf(omega_type,type,(
omega: \$i )).

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

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

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(notinself2_type,type,(
notinself2: \$o )).

thf(notinself2,definition,
( notinself2
= ( ! [A: \$i,B: \$i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ) ) )).

thf(omegaS_type,type,(
omegaS: \$i > \$i )).

thf(omegaS,definition,
( omegaS
= ( ^ [Xx: \$i] :
( setadjoin @ Xx @ Xx ) ) )).

thf(peanoSinj,conjecture,
=> ( in__Cong
=> ( notinself2
=> ! [Xx: \$i] :
( ( in @ Xx @ omega )
=> ! [Xy: \$i] :
( ( in @ Xy @ omega )
=> ( ( ( omegaS @ Xx )
= ( omegaS @ Xy ) )
=> ( Xx = Xy ) ) ) ) ) ) ) )).

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