SET007 Axioms: SET007+31.ax
%------------------------------------------------------------------------------
% File : SET007+31 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Finite Sets
% 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 : finset_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 18 unt; 0 def)
% Number of atoms : 169 ( 10 equ)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 129 ( 17 ~; 2 |; 63 &)
% ( 4 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-2 aty)
% Number of functors : 28 ( 28 usr; 4 con; 0-8 aty)
% Number of variables : 115 ( 109 !; 6 ?)
% 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(rc1_finset_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_finset_1(A) ) ).
fof(rc2_finset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_ordinal1(B)
& v2_ordinal1(B)
& v3_ordinal1(B)
& v4_ordinal2(B)
& v1_finset_1(B) ) ).
fof(fc1_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(k1_tarski(A))
& v1_finset_1(k1_tarski(A)) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(fc2_finset_1,axiom,
! [A,B] :
( ~ v1_xboole_0(k2_tarski(A,B))
& v1_finset_1(k2_tarski(A,B)) ) ).
fof(fc3_finset_1,axiom,
! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) ).
fof(fc4_finset_1,axiom,
! [A,B,C,D] : v1_finset_1(k2_enumset1(A,B,C,D)) ).
fof(fc5_finset_1,axiom,
! [A,B,C,D,E] : v1_finset_1(k3_enumset1(A,B,C,D,E)) ).
fof(fc6_finset_1,axiom,
! [A,B,C,D,E,F] : v1_finset_1(k4_enumset1(A,B,C,D,E,F)) ).
fof(fc7_finset_1,axiom,
! [A,B,C,D,E,F,G] : v1_finset_1(k5_enumset1(A,B,C,D,E,F,G)) ).
fof(fc8_finset_1,axiom,
! [A,B,C,D,E,F,G,H] : v1_finset_1(k6_enumset1(A,B,C,D,E,F,G,H)) ).
fof(cc2_finset_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_finset_1(B) ) ) ).
fof(fc9_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_xboole_0(A,B)) ) ).
fof(fc10_finset_1,axiom,
! [A,B] :
( v1_finset_1(B)
=> v1_finset_1(k3_xboole_0(A,B)) ) ).
fof(fc11_finset_1,axiom,
! [A,B] :
( v1_finset_1(A)
=> v1_finset_1(k3_xboole_0(A,B)) ) ).
fof(fc12_finset_1,axiom,
! [A,B] :
( v1_finset_1(A)
=> v1_finset_1(k4_xboole_0(A,B)) ) ).
fof(fc13_finset_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k9_relat_1(A,B)) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_zfmisc_1(A,B)) ) ).
fof(fc15_finset_1,axiom,
! [A,B,C] :
( ( v1_finset_1(A)
& v1_finset_1(B)
& v1_finset_1(C) )
=> v1_finset_1(k3_zfmisc_1(A,B,C)) ) ).
fof(fc16_finset_1,axiom,
! [A,B,C,D] :
( ( v1_finset_1(A)
& v1_finset_1(B)
& v1_finset_1(C)
& v1_finset_1(D) )
=> v1_finset_1(k4_zfmisc_1(A,B,C,D)) ) ).
fof(fc17_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k5_xboole_0(A,B)) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(d1_finset_1,axiom,
! [A] :
( v1_finset_1(A)
<=> ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& k2_relat_1(B) = A
& r2_hidden(k1_relat_1(B),k5_ordinal2) ) ) ).
fof(t1_finset_1,axiom,
$true ).
fof(t2_finset_1,axiom,
$true ).
fof(t3_finset_1,axiom,
$true ).
fof(t4_finset_1,axiom,
$true ).
fof(t5_finset_1,axiom,
$true ).
fof(t6_finset_1,axiom,
$true ).
fof(t7_finset_1,axiom,
$true ).
fof(t8_finset_1,axiom,
$true ).
fof(t9_finset_1,axiom,
$true ).
fof(t10_finset_1,axiom,
$true ).
fof(t11_finset_1,axiom,
$true ).
fof(t12_finset_1,axiom,
$true ).
fof(t13_finset_1,axiom,
! [A,B] :
( ( r1_tarski(A,B)
& v1_finset_1(B) )
=> v1_finset_1(A) ) ).
fof(t14_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_xboole_0(A,B)) ) ).
fof(t15_finset_1,axiom,
! [A,B] :
( v1_finset_1(A)
=> v1_finset_1(k3_xboole_0(A,B)) ) ).
fof(t16_finset_1,axiom,
! [A,B] :
( v1_finset_1(A)
=> v1_finset_1(k4_xboole_0(A,B)) ) ).
fof(t17_finset_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_finset_1(A)
=> v1_finset_1(k9_relat_1(B,A)) ) ) ).
fof(t18_finset_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( B != k1_xboole_0
& ! [C] :
~ ( r2_hidden(C,B)
& ! [D] :
( ( r2_hidden(D,B)
& r1_tarski(C,D) )
=> D = C ) ) ) ) ) ).
fof(t19_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_zfmisc_1(A,B)) ) ).
fof(t20_finset_1,axiom,
! [A,B,C] :
( ( v1_finset_1(A)
& v1_finset_1(B)
& v1_finset_1(C) )
=> v1_finset_1(k3_zfmisc_1(A,B,C)) ) ).
fof(t21_finset_1,axiom,
! [A,B,C,D] :
( ( v1_finset_1(A)
& v1_finset_1(B)
& v1_finset_1(C)
& v1_finset_1(D) )
=> v1_finset_1(k4_zfmisc_1(A,B,C,D)) ) ).
fof(t22_finset_1,axiom,
! [A,B] :
( v1_finset_1(k2_zfmisc_1(B,A))
=> ( A = k1_xboole_0
| v1_finset_1(B) ) ) ).
fof(t23_finset_1,axiom,
! [A,B] :
( v1_finset_1(k2_zfmisc_1(A,B))
=> ( A = k1_xboole_0
| v1_finset_1(B) ) ) ).
fof(t24_finset_1,axiom,
! [A] :
( v1_finset_1(A)
<=> v1_finset_1(k1_zfmisc_1(A)) ) ).
fof(t25_finset_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ! [B] :
( r2_hidden(B,A)
=> v1_finset_1(B) ) )
<=> v1_finset_1(k3_tarski(A)) ) ).
fof(t26_finset_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_finset_1(k1_relat_1(A))
=> v1_finset_1(k2_relat_1(A)) ) ) ).
fof(t27_finset_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( r1_tarski(A,k2_relat_1(B))
& v1_finset_1(k10_relat_1(B,A)) )
=> v1_finset_1(A) ) ) ).
fof(t28_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k5_xboole_0(A,B)) ) ).
fof(t29_finset_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_finset_1(k1_relat_1(A))
<=> v1_finset_1(A) ) ) ).
fof(t30_finset_1,axiom,
! [A] :
~ ( v1_finset_1(A)
& A != k1_xboole_0
& v6_ordinal1(A)
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
( r2_hidden(C,A)
=> r1_tarski(B,C) ) ) ) ).
fof(t31_finset_1,axiom,
! [A] :
~ ( v1_finset_1(A)
& A != k1_xboole_0
& v6_ordinal1(A)
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
( r2_hidden(C,A)
=> r1_tarski(C,B) ) ) ) ).
fof(s1_finset_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_finset_1
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,f1_s1_finset_1)
=> ( ( p1_s1_finset_1(B)
=> k1_funct_1(A,B) = f2_s1_finset_1(B) )
& ( ~ p1_s1_finset_1(B)
=> k1_funct_1(A,B) = f3_s1_finset_1(B) ) ) ) ) ) ).
fof(s2_finset_1,axiom,
( ( v1_finset_1(f1_s2_finset_1)
& p1_s2_finset_1(k1_xboole_0)
& ! [A,B] :
( ( r2_hidden(A,f1_s2_finset_1)
& r1_tarski(B,f1_s2_finset_1)
& p1_s2_finset_1(B) )
=> p1_s2_finset_1(k2_xboole_0(B,k1_tarski(A))) ) )
=> p1_s2_finset_1(f1_s2_finset_1) ) ).
%------------------------------------------------------------------------------