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Theorem fclsneii 22543
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 1130 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2 fclstop 22537 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
31, 2syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐽 ∈ Top)
4 simp2 1131 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5 eqid 2825 . . . . . 6 𝐽 = 𝐽
65neii1 21632 . . . . 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 21583 . . . 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 4215 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆))
115ntropn 21575 . . . 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 22542 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
141, 13syl 17 . . . . . . 7 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 𝐽)
1514snssd 4740 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ 𝐽)
165neiint 21630 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
173, 15, 7, 16syl3anc 1365 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
184, 17mpbid 233 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ ((int‘𝐽)‘𝑁))
19 snssg 4715 . . . . 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 258 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ ((int‘𝐽)‘𝑁))
22 simp3 1132 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑆𝐹)
23 fclsopni 22541 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
241, 12, 21, 22, 23syl13anc 1366 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
25 ssn0 4357 . 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 207  w3a 1081  wcel 2107  wne 3020  cin 3938  wss 3939  c0 4294  {csn 4563   cuni 4836  cfv 6351  (class class class)co 7151  Topctop 21419  intcnt 21543  neicnei 21623   fClus cfcls 22462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-fbas 20460  df-top 21420  df-topon 21437  df-cld 21545  df-ntr 21546  df-cls 21547  df-nei 21624  df-fil 22372  df-fcls 22467
This theorem is referenced by:  fclsnei  22545  fclsfnflim  22553
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