SET007 Axioms: SET007+326.ax
%------------------------------------------------------------------------------
% File : SET007+326 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Topological Space E^2_T. Simple Closed Curves
% 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 : topreal2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 10 ( 1 unt; 0 def)
% Number of atoms : 89 ( 13 equ)
% Maximal formula atoms : 21 ( 8 avg)
% Number of connectives : 103 ( 24 ~; 0 |; 51 &)
% ( 3 <=>; 25 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 11 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-4 aty)
% Number of functors : 10 ( 10 usr; 2 con; 0-3 aty)
% Number of variables : 30 ( 22 !; 8 ?)
% 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_topreal2,axiom,
( ~ v1_xboole_0(k2_topreal1)
& v1_topreal2(k2_topreal1) ) ).
fof(rc1_topreal2,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& v1_topreal2(A) ) ).
fof(cc1_topreal2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_topreal2(A)
=> ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2)) ) ) ) ).
fof(t1_topreal2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( A != B
& r2_hidden(A,k2_topreal1)
& r2_hidden(B,k2_topreal1)
& ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_topreal1(k15_euclid(np__2),A,B,C)
& r1_topreal1(k15_euclid(np__2),A,B,D)
& k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),C,D)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),C,D) = k2_struct_0(k15_euclid(np__2),A,B) ) ) ) ) ) ) ).
fof(t2_topreal2,axiom,
v6_compts_1(k2_topreal1,k15_euclid(np__2)) ).
fof(t3_topreal2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(k15_euclid(np__2),C)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),D)))
& m2_relset_1(E,u1_struct_0(k3_pre_topc(k15_euclid(np__2),C)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),D))) )
=> ( ( v3_tops_2(E,k3_pre_topc(k15_euclid(np__2),C),k3_pre_topc(k15_euclid(np__2),D))
& r1_topreal1(k15_euclid(np__2),A,B,C) )
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( F = k1_funct_1(E,A)
& G = k1_funct_1(E,B) )
=> r1_topreal1(k15_euclid(np__2),F,G,D) ) ) ) ) ) ) ) ) ) ).
fof(d1_topreal2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_topreal2(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k3_pre_topc(k15_euclid(np__2),k2_topreal1)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(B,u1_struct_0(k3_pre_topc(k15_euclid(np__2),k2_topreal1)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& v3_tops_2(B,k3_pre_topc(k15_euclid(np__2),k2_topreal1),k3_pre_topc(k15_euclid(np__2),A)) ) ) ) ).
fof(t4_topreal2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( v1_topreal2(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r2_hidden(B,A)
& r2_hidden(C,A) ) ) ) ) ) ).
fof(t5_topreal2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_topreal2(A)
<=> ( ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& B != C
& r2_hidden(B,A)
& r2_hidden(C,A) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r2_hidden(B,A)
& r2_hidden(C,A)
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,D)
& r1_topreal1(k15_euclid(np__2),B,C,E)
& A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),D,E)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) = k2_struct_0(k15_euclid(np__2),B,C) ) ) ) ) ) ) ) ) ) ).
fof(t6_topreal2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_topreal2(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& ? [D] :
( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ? [E] :
( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& B != C
& r2_hidden(B,A)
& r2_hidden(C,A)
& r1_topreal1(k15_euclid(np__2),B,C,D)
& r1_topreal1(k15_euclid(np__2),B,C,E)
& A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),D,E)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) = k2_struct_0(k15_euclid(np__2),B,C) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------