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Theorem fclselbas 23511
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 23505 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))
3 fclstopon 23507 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
42, 3mpbird 256 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
5 fclsopn 23509 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
64, 2, 5syl2anc 584 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
76ibi 266 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
87simpld 495 1 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  cin 3946  c0 4321   cuni 4907  cfv 6540  (class class class)co 7405  TopOnctopon 22403  Filcfil 23340   fClus cfcls 23431
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-fbas 20933  df-top 22387  df-topon 22404  df-cld 22514  df-ntr 22515  df-cls 22516  df-fil 23341  df-fcls 23436
This theorem is referenced by:  fclsneii  23512  fclsnei  23514  fclsfnflim  23522  flimfnfcls  23523  fcfelbas  23531  cnfcf  23537  cfilfcls  24782
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