Theorem fclselbas 24224

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) → 𝐴 ∈ 𝑋)

Proof of Theorem fclselbas

Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fclselbas.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	fclsfil 24218	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))
3		fclstopon 24220	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
4	2, 3	mpbird 260	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
5		fclsopn 24222	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
6	4, 2, 5	syl2anc 596	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
7	6	ibi 270	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
8	7	simpld 500	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   ∩ cin 3905  ∅c0 4286  ∪ cuni 4874  ‘cfv 6540  (class class class)co 7419  TopOnctopon 23117  Filcfil 24053   fClus cfcls 24144

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-fbas 21569  df-top 23101  df-topon 23118  df-cld 23226  df-ntr 23227  df-cls 23228  df-fil 24054  df-fcls 24149

This theorem is used by:  fclsneii  24225  fclsnei  24227  fclsfnflim  24235  flimfnfcls  24236  fcfelbas  24244  cnfcf  24250  cfilfcls  25484