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Theorem fclsneii 24026
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 1136 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2 fclstop 24020 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
31, 2syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐽 ∈ Top)
4 simp2 1137 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5 eqid 2736 . . . . . 6 𝐽 = 𝐽
65neii1 23115 . . . . 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 23066 . . . 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 4243 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆))
115ntropn 23058 . . . 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 24025 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
141, 13syl 17 . . . . . . 7 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 𝐽)
1514snssd 4808 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ 𝐽)
165neiint 23113 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
173, 15, 7, 16syl3anc 1372 . . . . 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 4782 . . . . 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 1138 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑆𝐹)
23 fclsopni 24024 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
241, 12, 21, 22, 23syl13anc 1373 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
25 ssn0 4403 . 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 2107  wne 2939  cin 3949  wss 3950  c0 4332  {csn 4625   cuni 4906  cfv 6560  (class class class)co 7432  Topctop 22900  intcnt 23026  neicnei 23106   fClus cfcls 23945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-top 22901  df-topon 22918  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-fil 23855  df-fcls 23950
This theorem is referenced by:  fclsnei  24028  fclsfnflim  24036
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