TPTP Problem File: SEU805^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU805^2 : TPTP v7.1.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : The Foundation Axiom
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.nonempty A -> (? X:i.in X A & binintersect X A = emptyset))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC307l [Bro08]

% Status   : Theorem
% Rating   : 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, 0.14 v6.1.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   12 (   0 unit;   7 type;   4 defn)
%            Number of atoms       :   52 (   7 equality;  23 variable)
%            Maximal formula depth :   11 (   5 average)
%            Number of connectives :   36 (   3   ~;   0   |;   4   &;  22   @)
%                                         (   0 <=>;   7  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   7   :;   0   =)
%            Number of variables   :   12 (   0 sgn;   5   !;   6   ?;   1   ^)
%                                         (  12   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

thf(emptyset_type,type,(
emptyset: \$i )).

thf(foundationAx_type,type,(
foundationAx: \$o )).

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

thf(nonempty_type,type,(
nonempty: \$i > \$o )).

thf(nonempty,definition,
( nonempty
= ( ^ [Xx: \$i] : ( Xx != emptyset ) ) )).

thf(nonemptyE1_type,type,(
nonemptyE1: \$o )).

thf(nonemptyE1,definition,
( nonemptyE1
= ( ! [A: \$i] :
( ( nonempty @ A )
=> ? [Xx: \$i] :
( in @ Xx @ A ) ) ) )).

thf(binintersect_type,type,(
binintersect: \$i > \$i > \$i )).

thf(disjointsetsI1_type,type,(
disjointsetsI1: \$o )).

thf(disjointsetsI1,definition,
( disjointsetsI1
= ( ! [A: \$i,B: \$i] :
( ~ ( ? [Xx: \$i] :
( ( in @ Xx @ A )
& ( in @ Xx @ B ) ) )
=> ( ( binintersect @ A @ B )
= emptyset ) ) ) )).

thf(foundation2,conjecture,
( foundationAx
=> ( nonemptyE1
=> ( disjointsetsI1
=> ! [A: \$i] :
( ( nonempty @ A )
=> ? [X: \$i] :
( ( in @ X @ A )
& ( ( binintersect @ X @ A )
= emptyset ) ) ) ) ) )).

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