## TPTP Problem File: SEV257^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV257^5 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Set Theory (Sets of sets)
% Problem  : TPS problem THM625
% Version  : Especial.
% English  : Empty sets are open in any topology.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0161 [Bro09]
%          : THM625 [TPS]

% Status   : Theorem
% Rating   : 0.17 v7.4.0, 0.11 v7.3.0, 0.20 v7.2.0, 0.12 v7.1.0, 0.14 v7.0.0, 0.12 v6.4.0, 0.14 v6.3.0, 0.17 v6.2.0, 0.33 v6.1.0, 0.17 v6.0.0, 0.00 v5.5.0, 0.20 v5.4.0, 0.50 v5.3.0, 0.75 v5.0.0, 0.50 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0
% Syntax   : Number of formulae    :    3 (   0 unit;   2 type;   0 defn)
%            Number of atoms       :   22 (   0 equality;  13 variable)
%            Maximal formula depth :   11 (   6 average)
%            Number of connectives :   21 (   0   ~;   0   |;   5   &;  12   @)
%                                         (   0 <=>;   4  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   2   :;   0   =)
%            Number of variables   :    9 (   2 sgn;   4   !;   1   ?;   4   ^)
%                                         (   9   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_NEQ_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
%          : Polymorphic definitions expanded.
%------------------------------------------------------------------------------
thf(a_type,type,(
a: \$tType )).

thf(cOPEN,type,(
cOPEN: ( a > \$o ) > \$o )).

thf(cTHM625_pme,conjecture,
( ( ( cOPEN
@ ^ [Xy: a] : \$true )
& ! [K: ( a > \$o ) > \$o] :
( ! [Xx: a > \$o] :
( ( K @ Xx )
=> ( cOPEN @ Xx ) )
=> ( cOPEN
@ ^ [Xx: a] :
? [S: a > \$o] :
( ( K @ S )
& ( S @ Xx ) ) ) )
& ! [Y: a > \$o,Z: a > \$o] :
( ( ( cOPEN @ Y )
& ( cOPEN @ Z ) )
=> ( cOPEN
@ ^ [Xx: a] :
( ( Y @ Xx )
& ( Z @ Xx ) ) ) ) )
=> ( cOPEN
@ ^ [Xy: a] : \$false ) )).

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