SET007 Axioms: SET007+852.ax
%------------------------------------------------------------------------------
% File : SET007+852 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Continuous Mappings between Topological Spaces
% Version : [Urb08] axioms.
% English : Continuous Mappings between Finite and One-Dimensional Finite
% Topological Spaces
% 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 : fintopo4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 31 ( 0 unt; 0 def)
% Number of atoms : 239 ( 23 equ)
% Maximal formula atoms : 20 ( 7 avg)
% Number of connectives : 250 ( 42 ~; 2 |; 80 &)
% ( 9 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-4 aty)
% Number of functors : 32 ( 32 usr; 5 con; 0-4 aty)
% Number of variables : 88 ( 88 !; 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(d1_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_fintopo4(A,B,C)
<=> ( r1_xboole_0(k8_fin_topo(A,B),C)
& r1_xboole_0(B,k8_fin_topo(A,C)) ) ) ) ) ) ).
fof(t1_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k7_fintopo3(A,B,C),k7_fintopo3(A,B,D)) ) ) ) ) ) ).
fof(t2_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k3_fintopo3(A,B,C),k3_fintopo3(A,B,D)) ) ) ) ) ) ).
fof(t3_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k9_fintopo3(A,B,D),k9_fintopo3(A,B,C)) ) ) ) ) ) ).
fof(t4_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k5_fintopo3(A,B,D),k5_fintopo3(A,B,C)) ) ) ) ) ) ).
fof(t5_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_fintopo4(A,B,C)
=> r1_fintopo4(A,C,B) ) ) ) ) ).
fof(t6_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_fintopo4(A,B,C)
=> r1_xboole_0(B,C) ) ) ) ) ).
fof(t7_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fin_topo(A)
=> ( r1_fintopo4(A,B,C)
<=> ( r1_xboole_0(k11_fin_topo(A,B),C)
& r1_xboole_0(B,k11_fin_topo(A,C)) ) ) ) ) ) ) ).
fof(t8_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_fin_topo(A)
& r1_xboole_0(k8_fin_topo(A,B),C) )
=> r1_xboole_0(B,k8_fin_topo(A,C)) ) ) ) ) ).
fof(t9_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_fin_topo(A)
& r1_xboole_0(B,k8_fin_topo(A,C)) )
=> r1_xboole_0(k8_fin_topo(A,B),C) ) ) ) ) ).
fof(t10_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fin_topo(A)
=> ( r1_fintopo4(A,B,C)
<=> r1_xboole_0(k8_fin_topo(A,B),C) ) ) ) ) ) ).
fof(t11_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fin_topo(A)
=> ( r1_fintopo4(A,B,C)
<=> r1_xboole_0(B,k8_fin_topo(A,C)) ) ) ) ) ) ).
fof(t12_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fin_topo(A)
=> ( v6_fin_topo(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B = k4_subset_1(u1_struct_0(A),C,D)
& r1_fintopo4(A,C,D)
& C != B
& D != B ) ) ) ) ) ) ) ).
fof(t13_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fin_topo(A)
=> ( v6_fin_topo(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C != k1_xboole_0
& k6_subset_1(u1_struct_0(A),B,C) != k1_xboole_0
& r1_tarski(C,B)
& r1_xboole_0(k8_fin_topo(A,C),k6_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ) ).
fof(d2_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_fintopo4(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( r2_hidden(E,u1_struct_0(A))
& F = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) )
=> r1_tarski(k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k10_fintopo3(A,np__0,E)),k10_fintopo3(B,D,F)) ) ) ) ) ) ) ) ) ).
fof(t14_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_fin_topo(B)
& l1_fin_topo(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( r2_fintopo4(A,B,D,np__0)
=> r2_fintopo4(A,B,D,C) ) ) ) ) ) ).
fof(t15_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_fin_topo(B)
& l1_fin_topo(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( r2_fintopo4(A,B,E,C)
& r1_xreal_0(C,D) )
=> r2_fintopo4(A,B,E,D) ) ) ) ) ) ) ).
fof(t16_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( r2_fintopo4(A,B,E,np__0)
& D = k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,C) )
=> r1_tarski(k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,k8_fin_topo(A,C)),k8_fin_topo(B,D)) ) ) ) ) ) ) ).
fof(t17_fintopo4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v6_fin_topo(C,A)
& r2_fintopo4(A,B,E,np__0)
& D = k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,C) )
=> v6_fin_topo(D,B) ) ) ) ) ) ) ).
fof(d3_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
& m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) )
=> ( B = k1_fintopo4(A)
<=> ( k1_relat_1(B) = k2_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(A))
=> k1_funct_1(B,C) = k1_enumset1(C,k1_limfunc1(k5_binarith(C,np__1),np__1),k1_rfunct_3(k1_nat_1(C,np__1),A)) ) ) ) ) ) ) ).
fof(d4_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k2_fintopo4(A) = g1_fin_topo(k2_finseq_1(A),k1_fintopo4(A)) ) ) ).
fof(t18_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> v2_fin_topo(k2_fintopo4(A)) ) ) ).
fof(t19_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> v3_fin_topo(k2_fintopo4(A)) ) ) ).
fof(d5_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
& m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) )
=> ( B = k3_fintopo4(A)
<=> ( k1_relat_1(B) = k2_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(A))
=> ( ~ ( ~ r1_xreal_0(C,np__1)
& ~ r1_xreal_0(A,C)
& k1_funct_1(B,C) != k1_enumset1(C,k5_binarith(C,np__1),k1_nat_1(C,np__1)) )
& ( C = np__1
=> ( r1_xreal_0(A,C)
| k1_funct_1(B,C) = k1_enumset1(C,A,k1_nat_1(C,np__1)) ) )
& ( C = A
=> ( r1_xreal_0(C,np__1)
| k1_funct_1(B,C) = k1_enumset1(C,k5_binarith(C,np__1),np__1) ) )
& ( ( C = np__1
& C = A )
=> k1_funct_1(B,C) = k1_tarski(C) ) ) ) ) ) ) ) ) ).
fof(d6_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k4_fintopo4(A) = g1_fin_topo(k2_finseq_1(A),k3_fintopo4(A)) ) ) ).
fof(t20_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> v2_fin_topo(k4_fintopo4(A)) ) ) ).
fof(t21_fintopo4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> v3_fin_topo(k4_fintopo4(A)) ) ) ).
fof(dt_k1_fintopo4,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k1_fintopo4(A))
& v1_funct_2(k1_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
& m2_relset_1(k1_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) ) ) ).
fof(dt_k2_fintopo4,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k2_fintopo4(A))
& l1_fin_topo(k2_fintopo4(A)) ) ) ).
fof(dt_k3_fintopo4,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k3_fintopo4(A))
& v1_funct_2(k3_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
& m2_relset_1(k3_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) ) ) ).
fof(dt_k4_fintopo4,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k4_fintopo4(A))
& l1_fin_topo(k4_fintopo4(A)) ) ) ).
%------------------------------------------------------------------------------