SET007 Axioms: SET007+712.ax
%------------------------------------------------------------------------------
% File : SET007+712 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hierarchies and Classifications of 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 : taxonom2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 6 unt; 0 def)
% Number of atoms : 241 ( 14 equ)
% Maximal formula atoms : 20 ( 6 avg)
% Number of connectives : 258 ( 52 ~; 5 |; 111 &)
% ( 10 <=>; 80 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 33 ( 32 usr; 0 prp; 1-3 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-3 aty)
% Number of variables : 109 ( 102 !; 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(rc1_taxonom2,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_taxonom2(A)
& v2_taxonom2(A) ) ).
fof(cc1_taxonom2,axiom,
! [A] :
( v1_realset1(A)
=> v3_taxonom2(A) ) ).
fof(rc2_taxonom2,axiom,
? [A] :
( ~ v1_realset1(A)
& v3_taxonom2(A) ) ).
fof(rc3_taxonom2,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v4_taxonom2(B) ) ).
fof(rc4_taxonom2,axiom,
! [A] :
? [B] :
( m1_taxonom2(B,A)
& v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v5_taxonom2(B,A) ) ).
fof(d1_taxonom2,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_taxonom2(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& r8_orders_1(u1_orders_2(A),B) ) ) ) ).
fof(d2_taxonom2,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_taxonom2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r1_orders_2(A,D,B)
& r1_orders_2(A,D,C) )
& ~ r1_orders_2(A,B,C)
& ~ r1_orders_2(A,C,B) ) ) ) ) ) ).
fof(t1_taxonom2,axiom,
! [A] :
( ~ v3_struct_0(k2_yellow_1(k1_tarski(k1_tarski(A))))
& v2_orders_2(k2_yellow_1(k1_tarski(k1_tarski(A))))
& v3_orders_2(k2_yellow_1(k1_tarski(k1_tarski(A))))
& v4_orders_2(k2_yellow_1(k1_tarski(k1_tarski(A))))
& v1_taxonom2(k2_yellow_1(k1_tarski(k1_tarski(A))))
& v2_taxonom2(k2_yellow_1(k1_tarski(k1_tarski(A)))) ) ).
fof(t2_taxonom2,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,D,E] :
( ( r2_hidden(E,k6_eqrel_1(A,B,C))
& r2_hidden(E,k6_eqrel_1(A,B,D)) )
=> k6_eqrel_1(A,B,C) = k6_eqrel_1(A,B,D) ) ) ) ).
fof(t3_taxonom2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C,D,E] :
( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(E,C)
& r2_hidden(E,D) )
=> C = D ) ) ) ).
fof(t4_taxonom2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( m1_taxonom1(B,A)
& r2_hidden(C,k3_tarski(B)) )
=> r1_tarski(C,A) ) ) ).
fof(t5_taxonom2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_taxonom1(B,A)
=> ( v2_orders_2(k2_yellow_1(k3_tarski(B)))
& v3_orders_2(k2_yellow_1(k3_tarski(B)))
& v4_orders_2(k2_yellow_1(k3_tarski(B)))
& v1_taxonom2(k2_yellow_1(k3_tarski(B)))
& v2_taxonom2(k2_yellow_1(k3_tarski(B)))
& l1_orders_2(k2_yellow_1(k3_tarski(B))) ) ) ) ).
fof(d3_taxonom2,axiom,
! [A] :
( v3_taxonom2(A)
<=> ! [B,C] :
~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& ~ r1_tarski(B,C)
& ~ r1_tarski(C,B)
& ~ r1_xboole_0(B,C) ) ) ).
fof(t6_taxonom2,axiom,
v3_taxonom2(k1_xboole_0) ).
fof(t7_taxonom2,axiom,
v3_taxonom2(k1_tarski(k1_xboole_0)) ).
fof(d4_taxonom2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( m1_taxonom2(B,A)
<=> v3_taxonom2(B) ) ) ).
fof(d5_taxonom2,axiom,
! [A] :
( v4_taxonom2(A)
<=> ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> ( B = C
| r1_xboole_0(B,C) ) ) ) ).
fof(t8_taxonom2,axiom,
v4_taxonom2(k1_xboole_0) ).
fof(t9_taxonom2,axiom,
v4_taxonom2(k1_tarski(k1_xboole_0)) ).
fof(t10_taxonom2,axiom,
! [A] : v4_taxonom2(k1_tarski(A)) ).
fof(d6_taxonom2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v5_taxonom2(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( ~ r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ~ ( r2_hidden(D,E)
& r2_hidden(E,B)
& r1_xboole_0(C,E) ) ) ) ) ) ) ) ) ).
fof(t11_taxonom2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B = k1_xboole_0
=> v5_taxonom2(B,A) ) ) ).
fof(d7_taxonom2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v6_taxonom2(B,A)
<=> ! [C] :
~ ( C != k1_xboole_0
& r1_tarski(C,B)
& v6_ordinal1(C)
& ! [D] :
~ ( r2_hidden(D,B)
& r1_tarski(D,k1_setfam_1(C)) ) ) ) ) ).
fof(t12_taxonom2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( ( v4_taxonom2(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ! [D] :
( r2_hidden(D,C)
=> ( r1_xboole_0(B,D)
& A != k1_xboole_0 ) )
=> ( v4_taxonom2(k2_xboole_0(C,k1_tarski(B)))
& m1_subset_1(k2_xboole_0(C,k1_tarski(B)),k1_zfmisc_1(k1_zfmisc_1(A)))
& ~ ( B != k1_xboole_0
& k3_tarski(k2_xboole_0(C,k1_tarski(B))) = k5_setfam_1(A,C) ) ) ) ) ) ).
fof(d8_taxonom2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v7_taxonom2(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r1_tarski(C,D)
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( r1_tarski(D,E)
& r2_hidden(E,B) )
=> E = A ) ) ) ) ) ) ) ) ).
fof(t13_taxonom2,axiom,
! [A,B] :
( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_taxonom2(B,A) )
=> ~ ( v7_taxonom2(B,A)
& ! [C] :
( m1_eqrel_1(C,A)
=> ~ r1_tarski(C,B) ) ) ) ).
fof(t14_taxonom2,axiom,
! [A,B] :
( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_taxonom2(B,A) )
=> ! [C] :
( ( v4_taxonom2(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r1_tarski(C,B)
& ! [D] :
( ( v4_taxonom2(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r1_tarski(D,B)
& r1_tarski(k5_setfam_1(A,C),k5_setfam_1(A,D)) )
=> C = D ) ) )
=> m1_eqrel_1(C,A) ) ) ) ).
fof(t15_taxonom2,axiom,
! [A,B] :
( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v5_taxonom2(B,A)
& m1_taxonom2(B,A) )
=> ( v6_taxonom2(B,A)
=> ( r2_hidden(k1_xboole_0,B)
| ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( ( v4_taxonom2(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r2_hidden(C,D)
& r1_tarski(D,B)
& ! [E] :
( ( v4_taxonom2(E)
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r2_hidden(C,E)
& r1_tarski(E,B)
& r1_tarski(k5_setfam_1(A,D),k5_setfam_1(A,E)) )
=> k5_setfam_1(A,D) = k5_setfam_1(A,E) ) ) )
=> m1_eqrel_1(D,A) ) ) ) ) ) ) ).
fof(t16_taxonom2,axiom,
! [A,B] :
( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v5_taxonom2(B,A)
& m1_taxonom2(B,A) )
=> ( v6_taxonom2(B,A)
=> ( r2_hidden(k1_xboole_0,B)
| ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( ( v4_taxonom2(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r2_hidden(C,D)
& r1_tarski(D,B)
& ! [E] :
( ( v4_taxonom2(E)
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r2_hidden(C,E)
& r1_tarski(E,B)
& r1_tarski(D,E) )
=> D = E ) ) )
=> m1_eqrel_1(D,A) ) ) ) ) ) ) ).
fof(t17_taxonom2,axiom,
! [A,B] :
( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v5_taxonom2(B,A)
& m1_taxonom2(B,A) )
=> ( v6_taxonom2(B,A)
=> ( r2_hidden(k1_xboole_0,B)
| ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_eqrel_1(D,A)
=> ~ ( r2_hidden(C,D)
& r1_tarski(D,B) ) ) ) ) ) ) ) ).
fof(t18_taxonom2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( ( r2_hidden(D,C)
& r1_tarski(B,D) )
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( ( r2_hidden(B,E)
& ! [F] :
~ ( r2_hidden(F,E)
& ~ r1_tarski(F,D)
& ~ r1_tarski(D,F)
& ~ r1_xboole_0(D,F) ) )
=> ! [F] :
( ! [G] :
( r2_hidden(G,F)
<=> ( r2_hidden(G,E)
& r1_tarski(G,D) ) )
=> ( m1_eqrel_1(k2_xboole_0(F,k4_xboole_0(C,k1_tarski(D))),A)
& r1_setfam_1(k2_xboole_0(F,k4_xboole_0(C,k1_tarski(D))),C) ) ) ) ) ) ) ) ) ).
fof(t19_taxonom2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r1_tarski(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( ! [D] :
~ ( r2_hidden(D,C)
& ~ r1_tarski(D,B)
& ~ r1_tarski(B,D)
& ~ r1_xboole_0(B,D) )
=> ( ! [D] :
~ ( r2_hidden(D,C)
& r1_tarski(D,B) )
| ! [D] :
( ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(E,C)
& r1_xboole_0(E,B) ) )
=> ( m1_eqrel_1(k2_xboole_0(D,k1_tarski(B)),A)
& r1_setfam_1(C,k2_xboole_0(D,k1_tarski(B)))
& ! [E] :
( m1_eqrel_1(E,A)
=> ( r1_setfam_1(C,E)
=> ! [F] :
( ( r2_hidden(F,E)
& r1_tarski(B,F) )
=> r1_setfam_1(k2_xboole_0(D,k1_tarski(B)),E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_taxonom2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v5_taxonom2(B,A)
& m1_taxonom2(B,A) )
=> ~ ( v6_taxonom2(B,A)
& ~ r2_hidden(k1_xboole_0,B)
& ! [C] :
~ ( C != k1_xboole_0
& r1_tarski(C,k1_partit1(A))
& ! [D,E] :
~ ( r2_hidden(D,C)
& r2_hidden(E,C)
& ~ r1_setfam_1(D,E)
& ~ r1_setfam_1(E,D) )
& ! [D,E] :
~ ( r2_hidden(D,C)
& r2_hidden(E,C)
& ! [F] :
( r2_hidden(F,C)
=> ( r1_setfam_1(F,E)
& r1_setfam_1(D,F) ) ) ) )
& ! [C] :
( m1_taxonom1(C,A)
=> k3_tarski(C) != B ) ) ) ) ).
fof(dt_m1_taxonom2,axiom,
! [A,B] :
( m1_taxonom2(B,A)
=> m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(existence_m1_taxonom2,axiom,
! [A] :
? [B] : m1_taxonom2(B,A) ).
%------------------------------------------------------------------------------