## SET007 Axioms: SET007+9.ax

```%------------------------------------------------------------------------------
% File     : SET007+9 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Subsets - 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    : subset [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    5 (   0 unit)
%            Number of atoms       :   13 (   0 equality)
%            Maximal formula depth :    7 (   5 average)
%            Number of connectives :    9 (   1 ~  ;   1  |;   3  &)
%                                         (   1 <=>;   3 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    4 (   0 propositional; 1-2 arity)
%            Number of functors    :    1 (   0 constant; 1-1 arity)
%            Number of variables   :   12 (   0 singleton;  12 !;   0 ?)
%            Maximal term depth    :    2 (   1 average)

% 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_subset,axiom,(
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) )).

fof(t2_subset,axiom,(
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) )).

fof(t3_subset,axiom,(
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) )).

fof(t4_subset,axiom,(
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) )).

fof(t5_subset,axiom,(
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) )).
%------------------------------------------------------------------------------
```