Theorem fclsneii 21821

Description: A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclsneii	|- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( N i^i S ) =/= (/) )

Proof of Theorem fclsneii

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> A e. ( J fClus F ) )
2		fclstop 21815	. . . . 5 |- ( A e. ( J fClus F ) -> J e. Top )
3	1, 2	syl 17	. . . 4 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> J e. Top )
4		simp2 1062	. . . . 5 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> N e. ( ( nei ` J ) ` { A } ) )
5		eqid 2622	. . . . . 6 |- U. J = U. J
6	5	neii1 20910	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` { A } ) ) -> N C_ U. J )
7	3, 4, 6	syl2anc 693	. . . 4 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> N C_ U. J )
8	5	ntrss2 20861	. . . 4 |- ( ( J e. Top /\ N C_ U. J ) -> ( ( int ` J ) ` N ) C_ N )
9	3, 7, 8	syl2anc 693	. . 3 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( ( int ` J ) ` N ) C_ N )
10		ssrin 3838	. . 3 |- ( ( ( int ` J ) ` N ) C_ N -> ( ( ( int ` J ) ` N ) i^i S ) C_ ( N i^i S ) )
11	9, 10	syl 17	. 2 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( ( ( int ` J ) ` N ) i^i S ) C_ ( N i^i S ) )
12	5	ntropn 20853	. . . 4 |- ( ( J e. Top /\ N C_ U. J ) -> ( ( int ` J ) ` N ) e. J )
13	3, 7, 12	syl2anc 693	. . 3 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( ( int ` J ) ` N ) e. J )
14	5	fclselbas 21820	. . . . . . . 8 |- ( A e. ( J fClus F ) -> A e. U. J )
15	1, 14	syl 17	. . . . . . 7 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> A e. U. J )
16	15	snssd 4340	. . . . . 6 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> { A } C_ U. J )
17	5	neiint 20908	. . . . . 6 |- ( ( J e. Top /\ { A } C_ U. J /\ N C_ U. J ) -> ( N e. ( ( nei ` J ) ` { A } ) <-> { A } C_ ( ( int ` J ) ` N ) ) )
18	3, 16, 7, 17	syl3anc 1326	. . . . 5 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( N e. ( ( nei ` J ) ` { A } ) <-> { A } C_ ( ( int ` J ) ` N ) ) )
19	4, 18	mpbid 222	. . . 4 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> { A } C_ ( ( int ` J ) ` N ) )
20		snssg 4327	. . . . 5 |- ( A e. U. J -> ( A e. ( ( int ` J ) ` N ) <-> { A } C_ ( ( int ` J ) ` N ) ) )
21	15, 20	syl 17	. . . 4 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( A e. ( ( int ` J ) ` N ) <-> { A } C_ ( ( int ` J ) ` N ) ) )
22	19, 21	mpbird 247	. . 3 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> A e. ( ( int ` J ) ` N ) )
23		simp3 1063	. . 3 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> S e. F )
24		fclsopni 21819	. . 3 |- ( ( A e. ( J fClus F ) /\ ( ( ( int ` J ) ` N ) e. J /\ A e. ( ( int ` J ) ` N ) /\ S e. F ) ) -> ( ( ( int ` J ) ` N ) i^i S ) =/= (/) )
25	1, 13, 22, 23, 24	syl13anc 1328	. 2 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( ( ( int ` J ) ` N ) i^i S ) =/= (/) )
26		ssn0 3976	. 2 |- ( ( ( ( ( int ` J ) ` N ) i^i S ) C_ ( N i^i S ) /\ ( ( ( int ` J ) ` N ) i^i S ) =/= (/) ) -> ( N i^i S ) =/= (/) )
27	11, 25, 26	syl2anc 693	1 |- ( ( A e. ( J fClus F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. F ) -> ( N i^i S ) =/= (/) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   Topctop 20698   intcnt 20821   neicnei 20901   fClus cfcls 21740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650  df-fcls 21745

This theorem is referenced by:  fclsnei  21823  fclsfnflim  21831