%------------------------------------------------------------------------------
% File     : SEU534^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Bounded Quantifier Laws
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! phi:i>o.(? x:i.in x A & phi x) -> dsetconstr A
%            (^ x:i.phi x) = emptyset -> false)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC036l [Bro08]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.00 v6.1.0, 0.14 v6.0.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.60 v5.2.0, 0.40 v4.1.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :    8 (   2 unt;   5 typ;   2 def)
%            Number of atoms       :   12 (   3 equ;   0 cnn)
%            Maximal formula atoms :    5 (   4 avg)
%            Number of connectives :   24 (   0   ~;   0   |;   1   &;  16   @)
%                                         (   0 <=>;   7  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   4 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    7 (   7   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5 usr;   4 con; 0-2 aty)
%            Number of variables   :   10 (   2   ^;   7   !;   1   ?;  10   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=385
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

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

thf(dsetconstr_type,type,
    dsetconstr: $i > ( $i > $o ) > $i ).

thf(dsetconstrI_type,type,
    dsetconstrI: $o ).

thf(dsetconstrI,definition,
    ( dsetconstrI
    = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
          ( ( in @ Xx @ A )
         => ( ( Xphi @ Xx )
           => ( in @ Xx
              @ ( dsetconstr @ A
                @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).

thf(emptysetE_type,type,
    emptysetE: $o ).

thf(emptysetE,definition,
    ( emptysetE
    = ( ! [Xx: $i] :
          ( ( in @ Xx @ emptyset )
         => ! [Xphi: $o] : Xphi ) ) ) ).

thf(emptyE1,conjecture,
    ( dsetconstrI
   => ( emptysetE
     => ! [A: $i,Xphi: $i > $o] :
          ( ? [Xx: $i] :
              ( ( in @ Xx @ A )
              & ( Xphi @ Xx ) )
         => ( ( ( dsetconstr @ A
                @ ^ [Xx: $i] : ( Xphi @ Xx ) )
              = emptyset )
           => $false ) ) ) ) ).

%------------------------------------------------------------------------------