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Theorem fclsneii 21731
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 1059 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2 fclstop 21725 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
31, 2syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐽 ∈ Top)
4 simp2 1060 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5 eqid 2621 . . . . . 6 𝐽 = 𝐽
65neii1 20820 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 𝐽)
73, 4, 6syl2anc 692 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 𝐽)
85ntrss2 20771 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 𝐽) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
93, 7, 8syl2anc 692 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
10 ssrin 3816 . . 3 (((int‘𝐽)‘𝑁) ⊆ 𝑁 → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆))
119, 10syl 17 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆))
125ntropn 20763 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 𝐽) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
133, 7, 12syl2anc 692 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
145fclselbas 21730 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
151, 14syl 17 . . . . . . 7 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 𝐽)
1615snssd 4309 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ 𝐽)
175neiint 20818 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
183, 16, 7, 17syl3anc 1323 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
194, 18mpbid 222 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ ((int‘𝐽)‘𝑁))
20 snssg 4296 . . . . 5 (𝐴 𝐽 → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
2115, 20syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
2219, 21mpbird 247 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ ((int‘𝐽)‘𝑁))
23 simp3 1061 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑆𝐹)
24 fclsopni 21729 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
251, 13, 22, 23, 24syl13anc 1325 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
26 ssn0 3948 . 2 (((((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆) ∧ (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅) → (𝑁𝑆) ≠ ∅)
2711, 25, 26syl2anc 692 1 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036  wcel 1987  wne 2790  cin 3554  wss 3555  c0 3891  {csn 4148   cuni 4402  cfv 5847  (class class class)co 6604  Topctop 20617  intcnt 20731  neicnei 20811   fClus cfcls 21650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-fbas 19662  df-top 20621  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-fil 21560  df-fcls 21655
This theorem is referenced by:  fclsnei  21733  fclsfnflim  21741
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