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Theorem fclselbas 22628
 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 22622 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))
3 fclstopon 22624 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
42, 3mpbird 260 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
5 fclsopn 22626 . . . 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 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106   ∩ cin 3880  ∅c0 4243  ∪ cuni 4800  ‘cfv 6324  (class class class)co 7135  TopOnctopon 21522  Filcfil 22457   fClus cfcls 22548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-fbas 20091  df-top 21506  df-topon 21523  df-cld 21631  df-ntr 21632  df-cls 21633  df-fil 22458  df-fcls 22553 This theorem is referenced by:  fclsneii  22629  fclsnei  22631  fclsfnflim  22639  flimfnfcls  22640  fcfelbas  22648  cnfcf  22654  cfilfcls  23885
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