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Theorem flimelbas 21753
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 = 𝐽
Assertion
Ref Expression
flimelbas (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)

Proof of Theorem flimelbas
StepHypRef Expression
1 flimuni.1 . . . 4 𝑋 = 𝐽
21elflim2 21749 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simprbi 480 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
43simpld 475 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wss 3567  𝒫 cpw 4149  {csn 4168   cuni 4427  ran crn 5105  cfv 5876  (class class class)co 6635  Topctop 20679  neicnei 20882  Filcfil 21630   fLim cflim 21719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-top 20680  df-flim 21724
This theorem is referenced by:  flimfil  21754  flimss2  21757  flimss1  21758  flimclsi  21763  hausflimi  21765  flimsncls  21771  cnpflfi  21784  cnflf  21787  cnflf2  21788  flimcfil  23093
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