## TPTP Problem File: SEV461^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV461^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : EMPTY_UNIONS
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : EMPTY_UNIONS_.p [Kal16]

% Status   : Theorem
% Rating   : 0.00 v7.1.0
% Syntax   : Number of formulae    :   10 (   0 unit;   4 type;   0 defn)
%            Number of atoms       :   62 (   9 equality;  37 variable)
%            Maximal formula depth :   11 (   7 average)
%            Number of connectives :   37 (   1   ~;   0   |;   1   &;  34   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   20 (  20   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   4   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   22 (   0 sgn;  17   !;   1   ?;   0   ^)
%                                         (  22   :;   4  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_const_const/trivia/I',type,(
'const/trivia/I':
!>[A: \$tType] :
( A > A ) )).

thf('thf_const_const/sets/UNIONS',type,(
'const/sets/UNIONS':
!>[A: \$tType] :
( ( ( A > \$o ) > \$o ) > A > \$o ) )).

thf('thf_const_const/sets/IN',type,(
'const/sets/IN':
!>[A: \$tType] :
( A > ( A > \$o ) > \$o ) )).

thf('thf_const_const/sets/EMPTY',type,(
'const/sets/EMPTY':
!>[A: \$tType] :
( A > \$o ) )).

thf('thm/trivia/I_THM_',axiom,(
! [A: \$tType,A0: A] :
( ( 'const/trivia/I' @ A @ A0 )
= A0 ) )).

thf('thm/sets/NOT_IN_EMPTY_',axiom,(
! [A: \$tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) )).

thf('thm/sets/IN_',axiom,(
! [A: \$tType,P: A > \$o,A0: A] :
( ( 'const/sets/IN' @ A @ A0 @ P )
= ( P @ A0 ) ) )).

thf('thm/sets/IN_UNIONS_',axiom,(
! [A: \$tType,A0: ( A > \$o ) > \$o,A1: A] :
( ( 'const/sets/IN' @ A @ A1 @ ( 'const/sets/UNIONS' @ A @ A0 ) )
= ( ? [A2: A > \$o] :
( ( 'const/sets/IN' @ ( A > \$o ) @ A2 @ A0 )
& ( 'const/sets/IN' @ A @ A1 @ A2 ) ) ) ) )).

thf('thm/sets/EXTENSION_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A > \$o] :
( ( A0 = A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
= ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) )).

thf('thm/sets/EMPTY_UNIONS_',conjecture,(
! [A: \$tType,A0: ( A > \$o ) > \$o] :
( ( ( 'const/sets/UNIONS' @ A @ A0 )
= ( 'const/sets/EMPTY' @ A ) )
= ( ! [A1: A > \$o] :
( ( 'const/sets/IN' @ ( A > \$o ) @ A1 @ A0 )
=> ( A1
= ( 'const/sets/EMPTY' @ A ) ) ) ) ) )).

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