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Theorem flimelbas 17988
Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
flimuni.1  |-  X  = 
U. J
Assertion
Ref Expression
flimelbas  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )

Proof of Theorem flimelbas
StepHypRef Expression
1 flimuni.1 . . . 4  |-  X  = 
U. J
21elflim2 17984 . . 3  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) )
32simprbi 451 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
43simpld 446 1  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   ran crn 4870   ` cfv 5445  (class class class)co 6072   Topctop 16946   neicnei 17149   Filcfil 17865    fLim cflim 17954
This theorem is referenced by:  flimfil  17989  flimss2  17992  flimss1  17993  flimclsi  17998  hausflimi  18000  flimsncls  18006  cnpflfi  18019  cnflf  18022  cnflf2  18023  flimcfil  19254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-top 16951  df-flim 17959
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