SET007 Axioms: SET007+627.ax
%------------------------------------------------------------------------------
% File : SET007+627 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Predicate Calculus for Boolean Valued Functions. Part VI
% 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 : bvfunc14 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 4 unt; 0 def)
% Number of atoms : 359 ( 182 equ)
% Maximal formula atoms : 24 ( 10 avg)
% Number of connectives : 393 ( 67 ~; 82 |; 72 &)
% ( 0 <=>; 172 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 17 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 0 con; 1-5 aty)
% Number of variables : 218 ( 218 !; 0 ?)
% 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_bvfunc14,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> k22_bvfunc_1(A,B,k2_partit1(A,C,D)) = k5_subset_1(A,k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,D)) ) ) ) ) ).
fof(t2_bvfunc14,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
| k2_bvfunc_2(A,B) = k2_partit1(A,C,D) ) ) ) ) ) ) ).
fof(t3_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( B = k1_enumset1(C,D,E)
=> ( C = D
| D = E
| E = C
| k2_bvfunc_2(A,B) = k2_partit1(A,k2_partit1(A,C,D),E) ) ) ) ) ) ) ) ).
fof(t4_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( B = k1_enumset1(C,D,E)
=> ( C = D
| E = C
| k5_bvfunc_2(A,C,B) = k2_partit1(A,D,E) ) ) ) ) ) ) ) ).
fof(t5_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( B = k1_enumset1(C,D,E)
=> ( C = D
| D = E
| k5_bvfunc_2(A,D,B) = k2_partit1(A,E,C) ) ) ) ) ) ) ) ).
fof(t6_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( B = k1_enumset1(C,D,E)
=> ( D = E
| E = C
| k5_bvfunc_2(A,E,B) = k2_partit1(A,C,D) ) ) ) ) ) ) ) ).
fof(t7_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ( B = k2_enumset1(C,D,E,F)
=> ( C = D
| C = E
| C = F
| k5_bvfunc_2(A,C,B) = k2_partit1(A,k2_partit1(A,D,E),F) ) ) ) ) ) ) ) ) ).
fof(t8_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ( B = k2_enumset1(C,D,E,F)
=> ( C = D
| D = E
| D = F
| k5_bvfunc_2(A,D,B) = k2_partit1(A,k2_partit1(A,C,E),F) ) ) ) ) ) ) ) ) ).
fof(t9_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ( B = k2_enumset1(C,D,E,F)
=> ( C = E
| D = E
| E = F
| k5_bvfunc_2(A,E,B) = k2_partit1(A,k2_partit1(A,C,D),F) ) ) ) ) ) ) ) ) ).
fof(t10_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ( B = k2_enumset1(C,D,E,F)
=> ( C = F
| D = F
| E = F
| k5_bvfunc_2(A,F,B) = k2_partit1(A,k2_partit1(A,C,E),D) ) ) ) ) ) ) ) ) ).
fof(t11_bvfunc14,axiom,
$true ).
fof(t12_bvfunc14,axiom,
$true ).
fof(t13_bvfunc14,axiom,
$true ).
fof(t14_bvfunc14,axiom,
! [A,B,C,D,E,F] : k1_relat_1(k1_funct_4(k1_funct_4(k3_cqc_lang(A,D),k3_cqc_lang(B,E)),k3_cqc_lang(C,F))) = k1_enumset1(A,B,C) ).
fof(t15_bvfunc14,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D,E] :
( B != C
=> k1_funct_1(k1_funct_4(k1_funct_4(A,k3_cqc_lang(B,D)),k3_cqc_lang(C,E)),B) = D ) ) ).
fof(t16_bvfunc14,axiom,
! [A,B,C,D,E,F] :
~ ( A != B
& C != A
& k1_funct_1(k1_funct_4(k1_funct_4(k3_cqc_lang(A,D),k3_cqc_lang(B,E)),k3_cqc_lang(C,F)),A) != D ) ).
fof(t17_bvfunc14,axiom,
! [A,B,C,D,E,F,G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( G = k1_funct_4(k1_funct_4(k3_cqc_lang(A,D),k3_cqc_lang(B,E)),k3_cqc_lang(C,F))
=> k2_relat_1(G) = k1_enumset1(k1_funct_1(G,A),k1_funct_1(G,B),k1_funct_1(G,C)) ) ) ).
fof(t18_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ! [H,I,J,K] :
( ( B = k2_enumset1(C,D,E,F)
& G = k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(D,I),k3_cqc_lang(E,J)),k3_cqc_lang(F,K)),k3_cqc_lang(C,H)) )
=> ( C = D
| C = E
| C = F
| D = E
| D = F
| E = F
| ( k1_funct_1(G,D) = I
& k1_funct_1(G,E) = J
& k1_funct_1(G,F) = K ) ) ) ) ) ) ) ) ) ) ).
fof(t19_bvfunc14,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F,G,H,I] :
( E = k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,G),k3_cqc_lang(C,H)),k3_cqc_lang(D,I)),k3_cqc_lang(A,F))
=> k1_relat_1(E) = k2_enumset1(A,B,C,D) ) ) ).
fof(t20_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ! [H,I,J,K] :
( ( B = k2_enumset1(C,D,E,F)
& G = k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(D,I),k3_cqc_lang(E,J)),k3_cqc_lang(F,K)),k3_cqc_lang(C,H)) )
=> k2_relat_1(G) = k2_enumset1(k1_funct_1(G,C),k1_funct_1(G,D),k1_funct_1(G,E),k1_funct_1(G,F)) ) ) ) ) ) ) ) ) ).
fof(t21_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,A)
=> ! [I] :
( ( v1_relat_1(I)
& v1_funct_1(I) )
=> ~ ( v2_bvfunc_2(B,A)
& B = k2_enumset1(C,D,E,F)
& C != D
& C != E
& C != F
& D != E
& D != F
& E != F
& r1_xboole_0(k22_bvfunc_1(A,H,k2_partit1(A,k2_partit1(A,D,E),F)),k22_bvfunc_1(A,G,C)) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,A)
=> ~ ( v2_bvfunc_2(B,A)
& B = k2_enumset1(C,D,E,F)
& C != D
& C != E
& C != F
& D != E
& D != F
& E != F
& k22_bvfunc_1(A,G,k2_partit1(A,E,F)) = k22_bvfunc_1(A,H,k2_partit1(A,E,F))
& r1_xboole_0(k22_bvfunc_1(A,H,k5_bvfunc_2(A,C,B)),k22_bvfunc_1(A,G,k5_bvfunc_2(A,D,B))) ) ) ) ) ) ) ) ) ) ).
fof(t23_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ~ ( v2_bvfunc_2(B,A)
& B = k1_enumset1(C,D,E)
& C != D
& D != E
& E != C
& k22_bvfunc_1(A,F,E) = k22_bvfunc_1(A,G,E)
& r1_xboole_0(k22_bvfunc_1(A,G,k5_bvfunc_2(A,C,B)),k22_bvfunc_1(A,F,k5_bvfunc_2(A,D,B))) ) ) ) ) ) ) ) ) ).
fof(t24_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ( B = k3_enumset1(C,D,E,F,G)
=> ( C = D
| C = E
| C = F
| C = G
| k5_bvfunc_2(A,C,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,D,E),F),G) ) ) ) ) ) ) ) ) ) ).
fof(t25_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k3_enumset1(C,D,E,F,G) )
=> ( C = D
| C = E
| C = F
| C = G
| D = E
| D = F
| D = G
| E = F
| E = G
| F = G
| k5_bvfunc_2(A,D,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,C,E),F),G) ) ) ) ) ) ) ) ) ) ).
fof(t26_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k3_enumset1(C,D,E,F,G) )
=> ( C = D
| C = E
| C = F
| C = G
| D = E
| D = F
| D = G
| E = F
| E = G
| F = G
| k5_bvfunc_2(A,E,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),F),G) ) ) ) ) ) ) ) ) ) ).
fof(t27_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k3_enumset1(C,D,E,F,G) )
=> ( C = D
| C = E
| C = F
| C = G
| D = E
| D = F
| D = G
| E = F
| E = G
| F = G
| k5_bvfunc_2(A,F,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),E),G) ) ) ) ) ) ) ) ) ) ).
fof(t28_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k3_enumset1(C,D,E,F,G) )
=> ( C = D
| C = E
| C = F
| C = G
| D = E
| D = F
| D = G
| E = F
| E = G
| F = G
| k5_bvfunc_2(A,G,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),E),F) ) ) ) ) ) ) ) ) ) ).
fof(t29_bvfunc14,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G,H,I,J,K] :
( F = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,H),k3_cqc_lang(C,I)),k3_cqc_lang(D,J)),k3_cqc_lang(E,K)),k3_cqc_lang(A,G))
=> ( A = B
| A = C
| A = D
| A = E
| B = C
| B = D
| B = E
| C = D
| C = E
| D = E
| ( k1_funct_1(F,A) = G
& k1_funct_1(F,B) = H
& k1_funct_1(F,C) = I
& k1_funct_1(F,D) = J
& k1_funct_1(F,E) = K ) ) ) ) ).
fof(t30_bvfunc14,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G,H,I,J,K] :
( F = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,H),k3_cqc_lang(C,I)),k3_cqc_lang(D,J)),k3_cqc_lang(E,K)),k3_cqc_lang(A,G))
=> k1_relat_1(F) = k3_enumset1(A,B,C,D,E) ) ) ).
fof(t31_bvfunc14,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G,H,I,J,K] :
( F = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,H),k3_cqc_lang(C,I)),k3_cqc_lang(D,J)),k3_cqc_lang(E,K)),k3_cqc_lang(A,G))
=> k2_relat_1(F) = k3_enumset1(k1_funct_1(F,A),k1_funct_1(F,B),k1_funct_1(F,C),k1_funct_1(F,D),k1_funct_1(F,E)) ) ) ).
fof(t32_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ! [H] :
( m1_subset_1(H,A)
=> ! [I] :
( m1_subset_1(I,A)
=> ! [J] :
( ( v1_relat_1(J)
& v1_funct_1(J) )
=> ~ ( v2_bvfunc_2(B,A)
& B = k3_enumset1(C,D,E,F,G)
& C != D
& C != E
& C != F
& C != G
& D != E
& D != F
& D != G
& E != F
& E != G
& F != G
& r1_xboole_0(k22_bvfunc_1(A,I,k2_partit1(A,k2_partit1(A,k2_partit1(A,D,E),F),G)),k22_bvfunc_1(A,H,C)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t33_bvfunc14,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)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ! [G] :
( m1_eqrel_1(G,A)
=> ! [H] :
( m1_subset_1(H,A)
=> ! [I] :
( m1_subset_1(I,A)
=> ~ ( v2_bvfunc_2(B,A)
& B = k3_enumset1(C,D,E,F,G)
& C != D
& C != E
& C != F
& C != G
& D != E
& D != F
& D != G
& E != F
& E != G
& F != G
& k22_bvfunc_1(A,H,k2_partit1(A,k2_partit1(A,E,F),G)) = k22_bvfunc_1(A,I,k2_partit1(A,k2_partit1(A,E,F),G))
& r1_xboole_0(k22_bvfunc_1(A,I,k5_bvfunc_2(A,C,B)),k22_bvfunc_1(A,H,k5_bvfunc_2(A,D,B))) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------