## SET007 Axioms: SET007+671.ax

```%------------------------------------------------------------------------------
% File     : SET007+671 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Concept of Fuzzy Relation and Basic Properties of its Operation
% 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    : fuzzy_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   12 (   0 unit)
%            Number of atoms       :   50 (  13 equality)
%            Maximal formula depth :   10 (   7 average)
%            Number of connectives :   61 (  23 ~  ;   0  |;   7  &)
%                                         (   0 <=>;  31 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    7 (   0 propositional; 1-3 arity)
%            Number of functors    :   13 (   2 constant; 0-4 arity)
%            Number of variables   :   31 (   0 singleton;  31 !;   0 ?)
%            Maximal term depth    :    3 (   2 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(cc1_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> v1_funct_2(B,A,k1_numbers) ) ) )).

fof(d1_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k1_fuzzy_3(A,B) = k5_rfunct_1(k1_xboole_0,k2_zfmisc_1(A,B)) ) ) )).

fof(d2_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k2_fuzzy_3(A,B) = k5_rfunct_1(k2_zfmisc_1(A,B),k2_zfmisc_1(A,B)) ) ) )).

fof(t1_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k1_fuzzy_3(A,B) = k4_fuzzy_1(k2_zfmisc_1(A,B)) ) ) )).

fof(t2_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k2_fuzzy_3(A,B) = k5_fuzzy_1(k2_zfmisc_1(A,B)) ) ) )).

fof(t3_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( r1_xreal_0(k8_funct_2(k2_zfmisc_1(A,B),k1_numbers,k1_fuzzy_3(A,B),C),k8_funct_2(k2_zfmisc_1(A,B),k1_numbers,D,C))
& r1_xreal_0(k8_funct_2(k2_zfmisc_1(A,B),k1_numbers,D,C),k8_funct_2(k2_zfmisc_1(A,B),k1_numbers,k2_fuzzy_3(A,B),C)) ) ) ) ) ) )).

fof(t4_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ( k2_fuzzy_1(k2_zfmisc_1(A,B),C,k2_fuzzy_3(A,B)) = k2_fuzzy_3(A,B)
& k1_fuzzy_1(k2_zfmisc_1(A,B),C,k2_fuzzy_3(A,B)) = C
& k2_fuzzy_1(k2_zfmisc_1(A,B),C,k1_fuzzy_3(A,B)) = C
& k1_fuzzy_1(k2_zfmisc_1(A,B),C,k1_fuzzy_3(A,B)) = k1_fuzzy_3(A,B) ) ) ) ) )).

fof(t5_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( k3_fuzzy_1(k2_zfmisc_1(A,B),k1_fuzzy_3(A,B)) = k2_fuzzy_3(A,B)
& k3_fuzzy_1(k2_zfmisc_1(A,B),k2_fuzzy_3(A,B)) = k1_fuzzy_3(A,B) ) ) ) )).

fof(t6_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( k1_fuzzy_2(k2_zfmisc_1(A,B),C,D) = k1_fuzzy_3(A,B)
=> r1_fuzzy_1(C,D) ) ) ) ) ) )).

fof(t7_fuzzy_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( k1_fuzzy_1(k2_zfmisc_1(A,B),C,D) = k1_fuzzy_3(A,B)
=> k1_fuzzy_2(k2_zfmisc_1(A,B),C,D) = C ) ) ) ) ) )).

fof(dt_k1_fuzzy_3,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> m1_fuzzy_1(k1_fuzzy_3(A,B),k2_zfmisc_1(A,B)) ) )).

fof(dt_k2_fuzzy_3,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> m1_fuzzy_1(k2_fuzzy_3(A,B),k2_zfmisc_1(A,B)) ) )).
%------------------------------------------------------------------------------
```