%------------------------------------------------------------------------------
% File     : SEU807^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : The Foundation Axiom
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.in A B -> ~(in B A))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC309l [Bro08]

% Status   : Theorem
% Rating   : 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   14 (   5 unt;   8 typ;   5 def)
%            Number of atoms       :   31 (   9 equ;   0 cnn)
%            Maximal formula atoms :    7 (   5 avg)
%            Number of connectives :   49 (   2   ~;   1   |;   2   &;  32   @)
%                                         (   1 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   8 usr;   6 con; 0-2 aty)
%            Number of variables   :   18 (   0   ^;  15   !;   3   ?;  18   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=436
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

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

thf(setadjoin_type,type,
    setadjoin: $i > $i > $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(setadjoinIL_type,type,
    setadjoinIL: $o ).

thf(setadjoinIL,definition,
    ( setadjoinIL
    = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).

thf(setadjoinIR_type,type,
    setadjoinIR: $o ).

thf(setadjoinIR,definition,
    ( setadjoinIR
    = ( ! [Xx: $i,A: $i,Xy: $i] :
          ( ( in @ Xy @ A )
         => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).

thf(in__Cong_type,type,
    in__Cong: $o ).

thf(in__Cong,definition,
    ( in__Cong
    = ( ! [A: $i,B: $i] :
          ( ( A = B )
         => ! [Xx: $i,Xy: $i] :
              ( ( Xx = Xy )
             => ( ( in @ Xx @ A )
              <=> ( in @ Xy @ B ) ) ) ) ) ) ).

thf(upairset2E_type,type,
    upairset2E: $o ).

thf(upairset2E,definition,
    ( upairset2E
    = ( ! [Xx: $i,Xy: $i,Xz: $i] :
          ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
         => ( ( Xz = Xx )
            | ( Xz = Xy ) ) ) ) ) ).

thf(notinself2,conjecture,
    ( foundationAx
   => ( setadjoinIL
     => ( setadjoinIR
       => ( in__Cong
         => ( upairset2E
           => ! [A: $i,B: $i] :
                ( ( in @ A @ B )
               => ~ ( in @ B @ A ) ) ) ) ) ) ) ).

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