Theorem flimelbas 24342

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) → 𝐴 ∈ 𝑋)

Proof of Theorem flimelbas

Step	Hyp	Ref	Expression
1		flimuni.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	elflim2 24338	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simprbi 503	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
4	3	simpld 500	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  {csn 4588  ∪ cuni 4872  ran crn 5659  ‘cfv 6566  (class class class)co 7451  Topctop 23266  neicnei 23470  Filcfil 24219   fLim cflim 24308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-top 23267  df-flim 24313

This theorem is used by:  flimfil  24343  flimss2  24346  flimss1  24347  flimclsi  24352  hausflimi  24354  flimsncls  24360  cnpflfi  24373  cnflf  24376  cnflf2  24377  flimcfil  25690