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Theorem fclsneii 24041
Description: A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsneii ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁𝑆) ≠ ∅)

Proof of Theorem fclsneii
StepHypRef Expression
1 simp1 1135 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2 fclstop 24035 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
31, 2syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐽 ∈ Top)
4 simp2 1136 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5 eqid 2735 . . . . . 6 𝐽 = 𝐽
65neii1 23130 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 𝐽)
73, 4, 6syl2anc 584 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 𝐽)
85ntrss2 23081 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 𝐽) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
93, 7, 8syl2anc 584 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
109ssrind 4252 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆))
115ntropn 23073 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 𝐽) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
123, 7, 11syl2anc 584 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
135fclselbas 24040 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
141, 13syl 17 . . . . . . 7 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 𝐽)
1514snssd 4814 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ 𝐽)
165neiint 23128 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
173, 15, 7, 16syl3anc 1370 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
184, 17mpbid 232 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ ((int‘𝐽)‘𝑁))
19 snssg 4788 . . . . 5 (𝐴 𝐽 → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
2014, 19syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
2118, 20mpbird 257 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ ((int‘𝐽)‘𝑁))
22 simp3 1137 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑆𝐹)
23 fclsopni 24039 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
241, 12, 21, 22, 23syl13anc 1371 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
25 ssn0 4410 . 2 (((((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆) ∧ (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅) → (𝑁𝑆) ≠ ∅)
2610, 24, 25syl2anc 584 1 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086  wcel 2106  wne 2938  cin 3962  wss 3963  c0 4339  {csn 4631   cuni 4912  cfv 6563  (class class class)co 7431  Topctop 22915  intcnt 23041  neicnei 23121   fClus cfcls 23960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-fil 23870  df-fcls 23965
This theorem is referenced by:  fclsnei  24043  fclsfnflim  24051
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