## SET007 Axioms: SET007+3.ax

```%------------------------------------------------------------------------------
% File     : SET007+3 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Boolean Properties of Sets - Requirements
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : boole [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   5 unit)
%            Number of atoms       :   12 (   7 equality)
%            Maximal formula depth :    7 (   3 average)
%            Number of connectives :    7 (   3 ~  ;   0  |;   3  &)
%                                         (   0 <=>;   1 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    3 (   0 propositional; 1-2 arity)
%            Number of functors    :    5 (   1 constant; 0-2 arity)
%            Number of variables   :   10 (   0 singleton;  10 !;   0 ?)
%            Maximal term depth    :    2 (   1 average)
% SPC      :

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_boole,axiom,(
! [A] : k2_xboole_0(A,k1_xboole_0) = A )).

fof(t2_boole,axiom,(
! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 )).

fof(t3_boole,axiom,(
! [A] : k4_xboole_0(A,k1_xboole_0) = A )).

fof(t4_boole,axiom,(
! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 )).

fof(t5_boole,axiom,(
! [A] : k5_xboole_0(A,k1_xboole_0) = A )).

fof(t6_boole,axiom,(
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) )).

fof(t7_boole,axiom,(
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) )).

fof(t8_boole,axiom,(
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) )).
%------------------------------------------------------------------------------
```