SET007 Axioms: SET007+508.ax
%------------------------------------------------------------------------------
% File : SET007+508 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Same Equivalents of Well-foundedness
% 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 : wellfnd1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 1 unt; 0 def)
% Number of atoms : 213 ( 13 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 200 ( 22 ~; 1 |; 84 &)
% ( 12 <=>; 81 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-4 aty)
% Number of functors : 32 ( 32 usr; 8 con; 0-4 aty)
% Number of variables : 86 ( 79 !; 7 ?)
% 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(fc1_wellfnd1,axiom,
! [A,B] :
( ~ v1_xboole_0(k4_partfun1(A,B))
& v1_fraenkel(k4_partfun1(A,B)) ) ).
fof(fc2_wellfnd1,axiom,
! [A] :
( ~ v1_xboole_0(k2_card_1(A))
& v1_ordinal1(k2_card_1(A))
& v2_ordinal1(k2_card_1(A))
& v3_ordinal1(k2_card_1(A))
& v1_card_1(k2_card_1(A)) ) ).
fof(rc1_wellfnd1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v1_card_5(A) ) ).
fof(rc2_wellfnd1,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_wellfnd1(A) ) ).
fof(rc3_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v2_wellfnd1(B,A) ) ) ).
fof(fc3_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v12_waybel_0(k2_wellfnd1(A),A)
& v2_wellfnd1(k2_wellfnd1(A),A) ) ) ).
fof(t1_wellfnd1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(B,C)
& r1_tarski(A,k1_relat_1(B)) )
=> k7_relat_1(B,A) = k7_relat_1(C,A) ) ) ) ).
fof(t2_wellfnd1,axiom,
! [A] :
( v1_fraenkel(A)
=> ( ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r1_partfun1(B,C) ) ) )
=> ( v1_relat_1(k3_tarski(A))
& v1_funct_1(k3_tarski(A)) ) ) ) ).
fof(t3_wellfnd1,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& v1_card_5(A) )
=> ! [B] :
( ( r1_tarski(B,A)
& r2_hidden(k1_card_1(B),A) )
=> r2_hidden(k7_ordinal2(B),A) ) ) ).
fof(t4_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] : r1_tarski(k1_wellord1(u1_orders_2(A),B),u1_struct_0(A)) ) ).
fof(d1_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v12_waybel_0(B,A)
<=> ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(k4_tarski(D,C),u1_orders_2(A)) )
=> r2_hidden(D,B) ) ) ) ) ).
fof(t5_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v12_waybel_0(B,A)
& r2_hidden(C,B) )
=> r1_tarski(k1_wellord1(u1_orders_2(A),C),B) ) ) ) ).
fof(t6_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v12_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( C = k2_xboole_0(B,k1_tarski(D))
& r1_tarski(k1_wellord1(u1_orders_2(A),D),B) )
=> v12_waybel_0(C,A) ) ) ) ) ).
fof(d2_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_wellfnd1(A)
<=> r1_wellord1(u1_orders_2(A),u1_struct_0(A)) ) ) ).
fof(d3_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_wellfnd1(B,A)
<=> r1_wellord1(u1_orders_2(A),B) ) ) ) ).
fof(t7_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v2_wellfnd1(k1_struct_0(A,B),A)
& m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(t8_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v2_wellfnd1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v2_wellfnd1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v12_waybel_0(B,A)
=> ( v2_wellfnd1(k4_subset_1(u1_struct_0(A),B,C),A)
& m1_subset_1(k4_subset_1(u1_struct_0(A),B,C),k1_zfmisc_1(u1_struct_0(A))) ) ) ) ) ) ).
fof(t9_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_wellfnd1(A)
<=> k2_wellfnd1(A) = u1_struct_0(A) ) ) ).
fof(t10_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_tarski(k1_wellord1(u1_orders_2(A),B),k2_wellfnd1(A))
=> r2_hidden(B,k2_wellfnd1(A)) ) ) ) ).
fof(t11_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_wellfnd1(A)
<=> ! [B] :
( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_tarski(k1_wellord1(u1_orders_2(A),C),B)
=> r2_hidden(C,B) ) )
=> r1_tarski(u1_struct_0(A),B) ) ) ) ).
fof(d5_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r1_wellfnd1(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k1_funct_1(D,E) = k1_funct_1(C,k4_tarski(E,k7_relat_1(D,k1_wellord1(u1_orders_2(A),E)))) ) ) ) ) ) ) ).
fof(t12_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_wellfnd1(A)
<=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),B)
& m2_relset_1(D,u1_struct_0(A),B)
& r1_wellfnd1(A,B,C,D) ) ) ) ) ) ).
fof(t13_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ~ v1_realset1(B)
=> ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),B)
& m2_relset_1(D,u1_struct_0(A),B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),B)
& m2_relset_1(E,u1_struct_0(A),B) )
=> ( ( r1_wellfnd1(A,B,C,D)
& r1_wellfnd1(A,B,C,E) )
=> D = E ) ) ) )
=> v1_wellfnd1(A) ) ) ) ).
fof(t14_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_wellfnd1(A)
& l1_orders_2(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),B)
& m2_relset_1(D,u1_struct_0(A),B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),B)
& m2_relset_1(E,u1_struct_0(A),B) )
=> ( ( r1_wellfnd1(A,B,C,D)
& r1_wellfnd1(A,B,C,E) )
=> D = E ) ) ) ) ) ) ).
fof(d6_wellfnd1,axiom,
$true ).
fof(d7_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v3_wellfnd1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) != k8_funct_2(k5_numbers,u1_struct_0(A),B,C)
& r2_hidden(k4_tarski(k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,u1_struct_0(A),B,C)),u1_orders_2(A)) ) ) ) ) ) ).
fof(t15_wellfnd1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_wellfnd1(A)
<=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ~ v3_wellfnd1(B,A) ) ) ) ).
fof(s1_wellfnd1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k4_rfunct_3(f1_s1_wellfnd1,f2_s1_wellfnd1)))
& ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,f1_s1_wellfnd1,f2_s1_wellfnd1) )
=> ( r2_hidden(B,A)
<=> p1_s1_wellfnd1(B) ) ) ) ).
fof(s2_wellfnd1,axiom,
( ( p1_s2_wellfnd1(f2_s2_wellfnd1)
& v1_wellfnd1(f1_s2_wellfnd1) )
=> ? [A] :
( m1_subset_1(A,u1_struct_0(f1_s2_wellfnd1))
& p1_s2_wellfnd1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s2_wellfnd1))
=> ~ ( A != B
& p1_s2_wellfnd1(B)
& r2_hidden(k4_tarski(B,A),u1_orders_2(f1_s2_wellfnd1)) ) ) ) ) ).
fof(s3_wellfnd1,axiom,
( ( ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s3_wellfnd1))
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s3_wellfnd1))
=> ( r2_hidden(k4_tarski(B,A),u1_orders_2(f1_s3_wellfnd1))
=> ( B = A
| p1_s3_wellfnd1(B) ) ) )
=> p1_s3_wellfnd1(A) ) )
& v1_wellfnd1(f1_s3_wellfnd1) )
=> ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s3_wellfnd1))
=> p1_s3_wellfnd1(A) ) ) ).
fof(dt_k1_wellfnd1,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& v1_funct_1(C)
& m1_relset_1(C,u1_struct_0(A),B) )
=> m1_subset_1(k1_wellfnd1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(redefinition_k1_wellfnd1,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& v1_funct_1(C)
& m1_relset_1(C,u1_struct_0(A),B) )
=> k1_wellfnd1(A,B,C) = k1_relat_1(C) ) ).
fof(dt_k2_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> m1_subset_1(k2_wellfnd1(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(d4_wellfnd1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = k2_wellfnd1(A)
<=> B = k3_tarski(a_1_0_wellfnd1(A)) ) ) ) ).
fof(fraenkel_a_1_0_wellfnd1,axiom,
! [A,B] :
( l1_orders_2(B)
=> ( r2_hidden(A,a_1_0_wellfnd1(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v2_wellfnd1(C,B)
& v12_waybel_0(C,B) ) ) ) ).
%------------------------------------------------------------------------------