SET007 Axioms: SET007+577.ax

%------------------------------------------------------------------------------
% File     : SET007+577 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Classes of Independent Partitions
% 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    : partit_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   2 unit)
%            Number of atoms       :  142 (   9 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  137 (  20 ~  ;   0  |;  31  &)
%                                         (   5 <=>;  81 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-4 arity)
%            Number of functors    :   18 (   1 constant; 0-6 arity)
%            Number of variables   :   84 (   0 singleton;  82 !;   2 ?)
%            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(d1_partit_2,axiom,(
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m2_relset_1(D,A,B)
=> ( r1_partit_2(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> ( r2_hidden(k4_tarski(E,F),C)
=> r2_hidden(k4_tarski(E,F),D) ) ) ) ) ) ) )).

fof(t1_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> k2_bvfunc_2(A,k1_subset_1(k1_bvfunc_2(A))) = k6_partit1(A) ) )).

fof(t2_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> r1_partit_2(A,A,k3_eqrel_1(A,B,C),k7_relset_1(A,A,A,A,B,C)) ) ) ) )).

fof(t3_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> r1_partit_2(A,A,B,k1_eqrel_1(A)) ) ) )).

fof(t4_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( k7_relset_1(A,A,A,A,k1_eqrel_1(A),B) = k1_eqrel_1(A)
& k7_relset_1(A,A,A,A,B,k1_eqrel_1(A)) = k1_eqrel_1(A) ) ) ) )).

fof(t5_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(k4_tarski(C,D),k4_partit1(A,B))
<=> r2_hidden(C,k22_bvfunc_1(A,D,B)) ) ) ) ) ) )).

fof(t6_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( k4_partit1(A,D) = k7_relset_1(A,A,A,A,k4_partit1(A,B),k4_partit1(A,C))
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( r2_hidden(E,k22_bvfunc_1(A,F,D))
<=> ? [G] :
( m1_subset_1(G,A)
& r2_hidden(E,k22_bvfunc_1(A,G,B))
& r2_hidden(G,k22_bvfunc_1(A,F,C)) ) ) ) ) ) ) ) ) ) )).

fof(t7_partit_2,axiom,(
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( r1_relat_2(A,C)
& r1_relat_2(B,C) )
=> r1_relat_2(k5_relat_1(A,B),C) ) ) ) )).

fof(t8_partit_2,axiom,(
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_relat_2(A,B)
=> r1_tarski(B,k3_relat_1(A)) ) ) )).

fof(t9_partit_2,axiom,(
! [A,B] :
( m2_relset_1(B,A,A)
=> ( r1_relat_2(B,A)
=> A = k3_relat_1(B) ) ) )).

fof(t10_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( k7_relset_1(A,A,A,A,B,C) = k7_relset_1(A,A,A,A,C,B)
=> ( v3_relat_2(k7_relset_1(A,A,A,A,B,C))
& v8_relat_2(k7_relset_1(A,A,A,A,B,C))
& v1_partfun1(k7_relset_1(A,A,A,A,B,C),A,A)
& m2_relset_1(k7_relset_1(A,A,A,A,B,C),A,A) ) ) ) ) ) )).

fof(t11_partit_2,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))
=> ( r1_bvfunc_1(A,B,C)
=> r1_bvfunc_1(A,k5_valuat_1(A,C),k5_valuat_1(A,B)) ) ) ) ) )).

fof(t12_partit_2,axiom,(
\$true )).

fof(t13_partit_2,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),k6_bvfunc_2(A,C,D,E)) ) ) ) ) ) ) )).

fof(t14_partit_2,axiom,(
\$true )).

fof(t15_partit_2,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,k7_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E)) ) ) ) ) ) ) )).

fof(t16_partit_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ( v2_bvfunc_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ( ( r1_tarski(C,B)
& r1_tarski(D,B) )
=> k7_relset_1(A,A,A,A,k4_partit1(A,k2_bvfunc_2(A,C)),k4_partit1(A,k2_bvfunc_2(A,D))) = k7_relset_1(A,A,A,A,k4_partit1(A,k2_bvfunc_2(A,D)),k4_partit1(A,k2_bvfunc_2(A,C))) ) ) ) ) ) ) )).

fof(t17_partit_2,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)
=> k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E) = k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,E),C,D) ) ) ) ) ) ) )).

fof(t18_partit_2,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)
=> k7_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(t19_partit_2,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),k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) )).

fof(dt_m1_partit_2,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
=> ! [C] :
( m1_partit_2(C,A,B)
=> m1_eqrel_1(C,A) ) ) )).

fof(existence_m1_partit_2,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
=> ? [C] : m1_partit_2(C,A,B) ) )).

fof(redefinition_m1_partit_2,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
=> ! [C] :
( m1_partit_2(C,A,B)
<=> m1_subset_1(C,B) ) ) )).

fof(reflexivity_r1_partit_2,axiom,(
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> r1_partit_2(A,B,C,C) ) )).

fof(redefinition_r1_partit_2,axiom,(
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> ( r1_partit_2(A,B,C,D)
<=> r1_tarski(C,D) ) ) )).
%------------------------------------------------------------------------------