MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimnei Structured version   Unicode version

Theorem flimnei 18000
Description: A filter contains all of the neighborhoods of its limit points. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
flimnei  |-  ( ( A  e.  ( J 
fLim  F )  /\  N  e.  ( ( nei `  J
) `  { A } ) )  ->  N  e.  F )

Proof of Theorem flimnei
StepHypRef Expression
1 flimneiss 17999 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
21sselda 3349 1  |-  ( ( A  e.  ( J 
fLim  F )  /\  N  e.  ( ( nei `  J
) `  { A } ) )  ->  N  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   {csn 3815   ` cfv 5455  (class class class)co 6082   neicnei 17162    fLim cflim 17967
This theorem is referenced by:  flimclsi  18011  hausflimlem  18012  flimsncls  18019  flimcfil  19267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-top 16964  df-flim 17972
  Copyright terms: Public domain W3C validator