## SET007 Axioms: SET007+30.ax

```%------------------------------------------------------------------------------
% File     : SET007+30 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Non-Negative Real Numbers. Part II
% 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    : arytm_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   32 (   1 unit)
%            Number of atoms       :  161 (  38 equality)
%            Maximal formula depth :   11 (   8 average)
%            Number of connectives :  145 (  16 ~  ;   4  |;  18  &)
%                                         (   3 <=>; 104 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    4 (   1 propositional; 0-2 arity)
%            Number of functors    :    7 (   2 constant; 0-2 arity)
%            Number of variables   :   80 (   0 singleton;  80 !;   0 ?)
%            Maximal term depth    :    4 (   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_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( k7_arytm_2(A,B) = B
=> A = k1_xboole_0 ) ) ) )).

fof(t2_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ( k8_arytm_2(A,B) = k1_xboole_0
& A != k1_xboole_0
& B != k1_xboole_0 ) ) ) )).

fof(t3_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(B,C) )
=> r1_arytm_2(A,C) ) ) ) ) )).

fof(t4_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(B,A) )
=> A = B ) ) ) )).

fof(t5_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& B = k1_xboole_0 )
=> A = k1_xboole_0 ) ) ) )).

fof(t6_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( A = k1_xboole_0
=> r1_arytm_2(A,B) ) ) ) )).

fof(t7_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
<=> r1_arytm_2(k7_arytm_2(A,C),k7_arytm_2(B,C)) ) ) ) ) )).

fof(t8_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> r1_arytm_2(k8_arytm_2(A,C),k8_arytm_2(B,C)) ) ) ) ) )).

fof(d1_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r1_arytm_2(B,A)
=> ( C = k1_arytm_1(A,B)
<=> k7_arytm_2(C,B) = A ) )
& ( ~ r1_arytm_2(B,A)
=> ( C = k1_arytm_1(A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ) )).

fof(t9_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ( ~ r1_arytm_2(A,B)
& k1_arytm_1(A,B) = k1_xboole_0 ) ) ) )).

fof(t10_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& k1_arytm_1(B,A) = k1_xboole_0 )
=> A = B ) ) ) )).

fof(t11_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> r1_arytm_2(k1_arytm_1(A,B),A) ) ) )).

fof(t12_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(A,C) )
=> k7_arytm_2(B,k1_arytm_1(C,A)) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) )).

fof(t13_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> k7_arytm_2(C,k1_arytm_1(B,A)) = k1_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) )).

fof(t14_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(C,A) )
=> k7_arytm_2(k1_arytm_1(B,A),C) = k1_arytm_1(B,k1_arytm_1(A,C)) ) ) ) ) )).

fof(t15_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(A,C) )
=> k7_arytm_2(k1_arytm_1(C,A),B) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) )).

fof(t16_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> r1_arytm_2(k1_arytm_1(C,B),k1_arytm_1(C,A)) ) ) ) ) )).

fof(t17_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> r1_arytm_2(k1_arytm_1(A,C),k1_arytm_1(B,C)) ) ) ) ) )).

fof(d2_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(B,A)
=> k2_arytm_1(A,B) = k1_arytm_1(A,B) )
& ( ~ r1_arytm_2(B,A)
=> k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,k1_arytm_1(B,A)) ) ) ) ) )).

fof(t18_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> k2_arytm_1(A,A) = k1_xboole_0 ) )).

fof(t19_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( A = k1_xboole_0
=> ( B = k1_xboole_0
| k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,B) ) ) ) ) )).

fof(t20_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> k7_arytm_2(C,k1_arytm_1(B,A)) = k2_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) )).

fof(t21_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ~ r1_arytm_2(A,B)
=> k2_arytm_1(C,k1_arytm_1(A,B)) = k2_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) )).

fof(t22_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> ( r1_arytm_2(A,C)
| k2_arytm_1(B,k1_arytm_1(A,C)) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ) )).

fof(t23_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ~ ( ~ r1_arytm_2(A,B)
& ~ r1_arytm_2(A,C)
& k2_arytm_1(B,k1_arytm_1(A,C)) != k2_arytm_1(C,k1_arytm_1(A,B)) ) ) ) ) )).

fof(t24_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> k2_arytm_1(B,k7_arytm_2(A,C)) = k2_arytm_1(k1_arytm_1(B,A),C) ) ) ) ) )).

fof(t25_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(C,B) )
=> k2_arytm_1(k1_arytm_1(B,C),A) = k2_arytm_1(k1_arytm_1(B,A),C) ) ) ) ) )).

fof(t26_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(A,B)
=> k8_arytm_2(C,k1_arytm_1(B,A)) = k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) ) ) ) ) )).

fof(t27_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ~ ( ~ r1_arytm_2(A,B)
& C != k1_xboole_0
& k4_tarski(k1_xboole_0,k8_arytm_2(C,k1_arytm_1(A,B))) != k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) ) ) ) ) )).

fof(t28_arytm_1,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( r1_arytm_2(B,A)
=> ( k1_arytm_1(A,B) = k1_xboole_0
| C = k1_xboole_0
| k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) = k4_tarski(k1_xboole_0,k8_arytm_2(C,k1_arytm_1(A,B))) ) ) ) ) ) )).

fof(dt_k1_arytm_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> m1_subset_1(k1_arytm_1(A,B),k2_arytm_2) ) )).

fof(dt_k2_arytm_1,axiom,(
\$true )).
%------------------------------------------------------------------------------
```