## SET007 Axioms: SET007+622.ax

```%------------------------------------------------------------------------------
% File     : SET007+622 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% 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]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   60 (   5 unit)
%            Number of atoms       :  264 (  47 equality)
%            Maximal formula depth :   15 (   7 average)
%            Number of connectives :  222 (  18 ~  ;   1  |;  50  &)
%                                         (   6 <=>; 147 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-3 arity)
%            Number of functors    :   41 (   6 constant; 0-5 arity)
%            Number of variables   :  149 (   0 singleton; 148 !;   1 ?)
%            Maximal term depth    :    7 (   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.
%------------------------------------------------------------------------------
! [A] :
( m1_subset_1(A,k5_numbers)

\$true )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k4_nat_1(A,B) = k6_xcmplx_0(B,np__1)
=> k4_nat_1(k1_nat_1(A,np__1),B) = np__0 ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& ~ r1_xreal_0(k6_xcmplx_0(A,np__1),k4_nat_1(B,A))
& k4_nat_1(k1_nat_1(B,np__1),A) != k1_nat_1(k4_nat_1(B,A),np__1) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( A != np__0
=> k4_nat_1(k4_nat_1(B,k2_nat_1(A,C)),C) = k4_nat_1(B,C) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& k4_nat_1(k1_nat_1(B,np__1),A) != np__0
& k4_nat_1(k1_nat_1(B,np__1),A) != k1_nat_1(k4_nat_1(B,A),np__1) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& B != np__0
& r1_xreal_0(A,k3_nat_1(k4_nat_1(C,k2_wsierp_1(A,B)),k2_wsierp_1(A,k5_binarith(B,np__1)))) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> r1_nat_1(k2_wsierp_1(C,A),k2_wsierp_1(C,B)) ) ) ) ) )).

! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_xreal_0(np__0,B)
=> ( r1_xreal_0(A,np__0)
| k6_int_1(k6_int_1(C,k3_xcmplx_0(A,B)),B) = k6_int_1(C,B) ) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_radix_1(A) = k4_power(np__2,A) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_radix_1(A) != np__0 ) )).

! [A] :
<=> A = np__0 ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
=> v1_int_1(B) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
& r1_xreal_0(k2_xcmplx_0(k4_xcmplx_0(k1_radix_1(A)),np__1),B) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)

\$true )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
=> ( r2_hidden(A,k2_finseq_1(B))
=> m2_subset_1(k1_funct_1(D,A),k6_wsierp_1,k3_radix_1(C)) ) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
=> ( ( r2_hidden(A,k2_finseq_1(C))
& ( A = np__0
=> k4_radix_1(A,B,C,D) = np__0 ) ) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
=> ( r2_hidden(A,k2_finseq_1(B))

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_wsierp_1,k4_finseq_2(np__1,k6_wsierp_1))
=> ( k4_finseq_4(k5_numbers,k6_wsierp_1,B,np__1) = A
=> B = k13_binarith(k1_numbers,A) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
=> ! [D] :
( m2_finseq_2(D,k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1))
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k6_wsierp_1,D,E) = k6_radix_1(E,B,A,C) ) ) ) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(B))
=> k4_radix_1(E,A,B,D) = k9_radix_1(E,A,C) ) ) ) ) ) ) ) )).

! [A] :
( v1_int_1(A)
=> ( ( ~ r1_xreal_0(A,np__2)
& ( ~ r1_xreal_0(k4_xcmplx_0(np__2),A)
& ( ( r1_xreal_0(A,np__2)
& r1_xreal_0(k4_xcmplx_0(np__2),A) )
=> k11_radix_1(A) = np__0 ) ) ) )).

! [A] :
( v1_int_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k12_radix_1(np__0,A) = np__0 ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_xreal_0(np__2,A)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
<=> ~ r1_xreal_0(k2_wsierp_1(k1_radix_1(C),A),B) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,A)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
=> ! [E] :
=> ( ( r2_hidden(B,k2_finseq_1(C))
& r1_xreal_0(np__2,A) )

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
=> ! [D] :
=> ! [E] :
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k2_finseq_1(A))
=> k4_radix_1(F,B,A,E) = k13_radix_1(B,F,A,C,D) ) ) ) ) ) ) ) ) )).

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,A)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,B)

! [A] :
( m1_subset_1(A,k5_numbers)

\$true )).

! [A] :
( m1_subset_1(A,k5_numbers)

! [A] :
( m1_subset_1(A,k5_numbers)

! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)

! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)

! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)

! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)

! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)

! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )

! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )

! [A] :
( v1_int_1(A)

! [A,B] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers) )

! [A,B,C,D,E] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)

! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)

! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)