SET007 Axioms: SET007+256.ax
%------------------------------------------------------------------------------
% File : SET007+256 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Locally Connected Spaces
% 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 : connsp_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 7 unt; 0 def)
% Number of atoms : 355 ( 17 equ)
% Maximal formula atoms : 23 ( 7 avg)
% Number of connectives : 378 ( 68 ~; 0 |; 154 &)
% ( 17 <=>; 139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-3 aty)
% Number of variables : 130 ( 126 !; 4 ?)
% 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_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_connsp_2(C,A,B)
<=> r2_hidden(B,k1_tops_1(A,C)) ) ) ) ) ).
fof(d2_connsp_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(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)))
=> ( m2_connsp_2(C,A,B)
<=> r1_tarski(B,k1_tops_1(A,C)) ) ) ) ) ).
fof(t1_connsp_2,axiom,
$true ).
fof(t2_connsp_2,axiom,
$true ).
fof(t3_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(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)))
=> ( ( m1_connsp_2(C,A,B)
& m1_connsp_2(D,A,B) )
=> m1_connsp_2(k4_subset_1(u1_struct_0(A),C,D),A,B) ) ) ) ) ) ).
fof(t4_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(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)))
=> ( ( m1_connsp_2(C,A,B)
& m1_connsp_2(D,A,B) )
<=> m1_connsp_2(k5_subset_1(u1_struct_0(A),C,D),A,B) ) ) ) ) ) ).
fof(t5_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( v3_pre_topc(B,A)
& r2_hidden(C,B) )
=> m1_connsp_2(B,A,C) ) ) ) ) ).
fof(t6_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( m1_connsp_2(B,A,C)
=> r2_hidden(C,B) ) ) ) ) ).
fof(t7_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( m1_connsp_2(C,A,B)
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( m1_connsp_2(D,A,B)
& v3_pre_topc(D,A)
& r1_tarski(D,C) ) ) ) ) ) ) ).
fof(t8_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_connsp_2(C,A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(D,A)
& r1_tarski(D,C)
& r2_hidden(B,D) ) ) ) ) ) ).
fof(t9_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( m1_connsp_2(D,A,C)
& r1_tarski(D,B) ) ) ) ) ) ) ) ).
fof(t10_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m2_connsp_2(C,A,k1_struct_0(A,B))
<=> m1_connsp_2(C,A,B) ) ) ) ) ).
fof(t11_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_pre_topc(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( v3_pre_topc(B,A)
& m1_connsp_2(D,k3_pre_topc(A,B),C)
& D = E
& C = F )
=> m1_connsp_2(E,A,F) ) ) ) ) ) ) ) ).
fof(t12_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_pre_topc(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( m1_connsp_2(E,A,F)
& D = E
& C = F )
=> m1_connsp_2(D,k3_pre_topc(A,B),C) ) ) ) ) ) ) ) ).
fof(t13_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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)))
=> ( ( r3_connsp_1(A,B)
& r1_tarski(B,C) )
=> r4_connsp_1(A,C,B) ) ) ) ) ).
fof(t14_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( C = D
=> r1_tarski(k1_connsp_1(B,D),k1_connsp_1(A,C)) ) ) ) ) ) ).
fof(d3_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_connsp_2(A,B)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( m1_connsp_2(C,A,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( m1_connsp_2(D,A,B)
& v2_connsp_1(D,A)
& r1_tarski(D,C) ) ) ) ) ) ) ) ).
fof(d4_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_connsp_2(A,B) ) ) ) ).
fof(d5_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_connsp_2(A,B,C)
<=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B = D
& ! [E] :
( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,D)))
=> ~ ( E = C
& r1_connsp_2(k3_pre_topc(A,D),E) ) ) ) ) ) ) ) ) ).
fof(d6_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v2_connsp_2(B,A)
<=> v1_connsp_2(k3_pre_topc(A,B)) ) ) ) ).
fof(t15_connsp_2,axiom,
$true ).
fof(t16_connsp_2,axiom,
$true ).
fof(t17_connsp_2,axiom,
$true ).
fof(t18_connsp_2,axiom,
$true ).
fof(t19_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_connsp_2(A,B)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( m1_connsp_2(C,A,B)
& r4_connsp_1(A,C,D)
& r2_hidden(B,D) )
=> m1_connsp_2(D,A,B) ) ) ) ) ) ) ).
fof(t20_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_connsp_2(A,B)
<=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v3_pre_topc(C,A)
& r2_hidden(B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(k3_pre_topc(A,C)))
=> ~ ( D = B
& r2_hidden(B,k1_tops_1(k3_pre_topc(A,C),k1_connsp_1(k3_pre_topc(A,C),D))) ) ) ) ) ) ) ) ).
fof(t21_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r3_connsp_1(A,B)
=> v3_pre_topc(B,A) ) ) ) ) ).
fof(t22_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_connsp_2(A,B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( v3_pre_topc(C,A)
& r2_hidden(B,C) )
=> r2_connsp_2(A,C,B) ) ) ) ) ) ).
fof(t23_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_2(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v3_pre_topc(B,A)
=> v2_connsp_2(B,A) ) ) ) ) ).
fof(t24_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_2(A)
<=> ! [B] :
( ( ~ v1_xboole_0(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_pre_topc(B,A)
& r4_connsp_1(A,B,C) )
=> v3_pre_topc(C,A) ) ) ) ) ) ).
fof(t25_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_2(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B)))) )
=> ~ ( v2_connsp_1(C,k3_pre_topc(A,B))
& v3_pre_topc(C,k3_pre_topc(A,B))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v2_connsp_1(D,A)
& C = k5_subset_1(u1_struct_0(A),B,D) ) ) ) ) ) ) ) ).
fof(t26_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v5_compts_1(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)))
=> ~ ( B != k1_xboole_0
& C != k2_pre_topc(A)
& r1_tarski(B,C)
& v4_pre_topc(B,A)
& v3_pre_topc(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r1_tarski(B,D)
& r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) ) ) ) ).
fof(t27_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v1_connsp_2(A)
& v5_compts_1(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)))
=> ~ ( B != k1_xboole_0
& C != k1_xboole_0
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C)
& v2_connsp_1(B,A)
& v2_connsp_1(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v2_connsp_1(D,A)
& v2_connsp_1(E,A)
& v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ).
fof(t28_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,C)
<=> ( v3_pre_topc(D,A)
& v4_pre_topc(D,A)
& r2_hidden(B,D) ) ) )
& C = k1_xboole_0 ) ) ) ) ).
fof(d7_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k1_connsp_2(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,D)
<=> ( v3_pre_topc(E,A)
& v4_pre_topc(E,A)
& r2_hidden(B,E) ) ) )
& k6_setfam_1(u1_struct_0(A),D) = C ) ) ) ) ) ).
fof(t29_connsp_2,axiom,
$true ).
fof(t30_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(B,k1_connsp_2(A,B)) ) ) ).
fof(t31_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(C,A)
& v4_pre_topc(C,A)
& r2_hidden(B,C)
& r1_tarski(C,k1_connsp_2(A,B)) )
=> C = k1_connsp_2(A,B) ) ) ) ) ).
fof(t32_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v4_pre_topc(k1_connsp_2(A,B),A) ) ) ).
fof(t33_connsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(k1_connsp_1(A,B),k1_connsp_2(A,B)) ) ) ).
fof(dt_m1_connsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_connsp_2(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_connsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ? [C] : m1_connsp_2(C,A,B) ) ).
fof(dt_m2_connsp_2,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_connsp_2(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m2_connsp_2,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m2_connsp_2(C,A,B) ) ).
fof(dt_k1_connsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_connsp_2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
%------------------------------------------------------------------------------