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Theorem fclselbas 22913
Description: A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
fclselbas.1 𝑋 = 𝐽
Assertion
Ref Expression
fclselbas (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋)

Proof of Theorem fclselbas
Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fclselbas.1 . . . . . 6 𝑋 = 𝐽
21fclsfil 22907 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))
3 fclstopon 22909 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
42, 3mpbird 260 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
5 fclsopn 22911 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
64, 2, 5syl2anc 587 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
76ibi 270 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
87simpld 498 1 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  cin 3865  c0 4237   cuni 4819  cfv 6380  (class class class)co 7213  TopOnctopon 21807  Filcfil 22742   fClus cfcls 22833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-fbas 20360  df-top 21791  df-topon 21808  df-cld 21916  df-ntr 21917  df-cls 21918  df-fil 22743  df-fcls 22838
This theorem is referenced by:  fclsneii  22914  fclsnei  22916  fclsfnflim  22924  flimfnfcls  22925  fcfelbas  22933  cnfcf  22939  cfilfcls  24171
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