## SET007 Axioms: SET007+619.ax

```%------------------------------------------------------------------------------
% File     : SET007+619 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Predicate Calculus for Boolean Valued Functions. Part III
% 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    : bvfunc11 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   27 (   2 unit)
%            Number of atoms       :  165 (  13 equality)
%            Maximal formula depth :   16 (  10 average)
%            Number of connectives :  164 (  26 ~  ;   3  |;   5  &)
%                                         (   0 <=>; 130 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-4 arity)
%            Number of functors    :   15 (   1 constant; 0-4 arity)
%            Number of variables   :  119 (   0 singleton; 119 !;   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_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( r1_setfam_1(C,D)
=> r1_tarski(k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,D)) ) ) ) ) ) )).

fof(t2_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> r1_tarski(k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,k3_partit1(A,C,D))) ) ) ) ) )).

fof(t3_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> r1_tarski(k22_bvfunc_1(A,B,k2_partit1(A,C,D)),k22_bvfunc_1(A,B,C)) ) ) ) ) )).

fof(t4_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r1_tarski(k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,k6_partit1(A)))
& r1_tarski(k22_bvfunc_1(A,B,k3_pua2mss1(A)),k22_bvfunc_1(A,B,C)) ) ) ) ) )).

fof(t5_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k2_tarski(C,D) )
=> ( C = D
| ! [E,F] :
~ ( r2_hidden(E,C)
& r2_hidden(F,D)
& r1_xboole_0(E,F) ) ) ) ) ) ) ) )).

fof(t6_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k2_tarski(C,D) )
=> ( C = D
| k2_bvfunc_2(A,B) = k2_partit1(A,C,D) ) ) ) ) ) ) )).

fof(t7_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( B = k2_tarski(C,D)
=> ( C = D
| k5_bvfunc_2(A,C,B) = D ) ) ) ) ) ) )).

fof(t8_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r1_bvfunc_1(A,B,C)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E)) ) ) ) ) ) ) )).

fof(t9_bvfunc11,axiom,(
\$true )).

fof(t10_bvfunc11,axiom,(
\$true )).

fof(t11_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) )).

fof(t12_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) )).

fof(t13_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) )).

fof(t14_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) )).

fof(t15_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k5_valuat_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D)) ) ) ) ) ) )).

fof(t16_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,k5_valuat_1(A,k6_bvfunc_2(A,B,C,D)),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D)) ) ) ) ) ) ) )).

fof(t17_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k5_valuat_1(A,k6_bvfunc_2(A,B,C,E)),C,D) ) ) ) ) ) ) )).

fof(t18_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k5_valuat_1(A,k6_bvfunc_2(A,B,C,D)),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D)) ) ) ) ) ) ) )).

fof(t19_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D) ) ) ) ) ) ) )).

fof(t20_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E)) ) ) ) ) ) ) )).

fof(t21_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k6_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E) ) ) ) ) ) )).

fof(t22_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> k5_valuat_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k6_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E) ) ) ) ) ) )).

fof(t23_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E) ) ) ) ) ) )).

fof(t24_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) )).

fof(t25_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) ) ) ) ) ) )).

fof(t26_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) ) ) ) ) ) )).

fof(t27_bvfunc11,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) ) ) ) ) ) )).
%------------------------------------------------------------------------------
```