## TPTP Axioms File: SET008^0.ax

```%------------------------------------------------------------------------------
% File     : SET008^0 : TPTP v7.5.0. Released v3.6.0.
% Domain   : Set Theory
% Axioms   : Basic set theory definitions
% Version  : [Ben08] axioms.
% English  :

% Refs     : [BS+05] Benzmueller et al. (2005), Can a Higher-Order and a Fi
%          : [BS+08] Benzmueller et al. (2007), Combined Reasoning by Autom
%          : [Ben08] Benzmueller (2008), Email to Geoff Sutcliffe
% Source   : [Ben08]
% Names    : Typed_Set [Ben08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   28 (   0 unit;  14 type;  14 defn)
%            Number of atoms       :  200 (  18 equality;  46 variable)
%            Maximal formula depth :    9 (   6 average)
%            Number of connectives :   36 (   5   ~;   3   |;   6   &;  21   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :   70 (  70   >;   0   *;   0   +)
%            Number of symbols     :   17 (  14   :;   0  :=)
%            Number of variables   :   35 (   1 sgn;   1   !;   2   ?;  32   ^)
%                                         (  35   :;   0  :=;   0  !>;   0  ?*)
% SPC      :

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

thf(in,definition,
( in
= ( ^ [X: \$i,M: \$i > \$o] :
( M @ X ) ) )).

thf(is_a_decl,type,(
is_a: \$i > ( \$i > \$o ) > \$o )).

thf(is_a,definition,
( is_a
= ( ^ [X: \$i,M: \$i > \$o] :
( M @ X ) ) )).

thf(emptyset_decl,type,(
emptyset: \$i > \$o )).

thf(emptyset,definition,
( emptyset
= ( ^ [X: \$i] : \$false ) )).

thf(unord_pair_decl,type,(
unord_pair: \$i > \$i > \$i > \$o )).

thf(unord_pair,definition,
( unord_pair
= ( ^ [X: \$i,Y: \$i,U: \$i] :
( ( U = X )
| ( U = Y ) ) ) )).

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

thf(singleton,definition,
( singleton
= ( ^ [X: \$i,U: \$i] : ( U = X ) ) )).

thf(union_decl,type,(
union: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(union,definition,
( union
= ( ^ [X: \$i > \$o,Y: \$i > \$o,U: \$i] :
( ( X @ U )
| ( Y @ U ) ) ) )).

thf(excl_union_decl,type,(
excl_union: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(excl_union,definition,
( excl_union
= ( ^ [X: \$i > \$o,Y: \$i > \$o,U: \$i] :
( ( ( X @ U )
& ~ ( Y @ U ) )
| ( ~ ( X @ U )
& ( Y @ U ) ) ) ) )).

thf(intersection_decl,type,(
intersection: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(intersection,definition,
( intersection
= ( ^ [X: \$i > \$o,Y: \$i > \$o,U: \$i] :
( ( X @ U )
& ( Y @ U ) ) ) )).

thf(setminus_decl,type,(
setminus: ( \$i > \$o ) > ( \$i > \$o ) > \$i > \$o )).

thf(setminus,definition,
( setminus
= ( ^ [X: \$i > \$o,Y: \$i > \$o,U: \$i] :
( ( X @ U )
& ~ ( Y @ U ) ) ) )).

thf(complement_decl,type,(
complement: ( \$i > \$o ) > \$i > \$o )).

thf(complement,definition,
( complement
= ( ^ [X: \$i > \$o,U: \$i] :
~ ( X @ U ) ) )).

thf(disjoint_decl,type,(
disjoint: ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(disjoint,definition,
( disjoint
= ( ^ [X: \$i > \$o,Y: \$i > \$o] :
( ( intersection @ X @ Y )
= emptyset ) ) )).

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

thf(subset,definition,
( subset
= ( ^ [X: \$i > \$o,Y: \$i > \$o] :
! [U: \$i] :
( ( X @ U )
=> ( Y @ U ) ) ) )).

thf(meets_decl,type,(
meets: ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(meets,definition,
( meets
= ( ^ [X: \$i > \$o,Y: \$i > \$o] :
? [U: \$i] :
( ( X @ U )
& ( Y @ U ) ) ) )).

thf(misses_decl,type,(
misses: ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(misses,definition,
( misses
= ( ^ [X: \$i > \$o,Y: \$i > \$o] :
~ ( ? [U: \$i] :
( ( X @ U )
& ( Y @ U ) ) ) ) )).

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