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Theorem fclsneii 22043
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 1131 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2 fclstop 22037 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
31, 2syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐽 ∈ Top)
4 simp2 1132 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5 eqid 2761 . . . . . 6 𝐽 = 𝐽
65neii1 21133 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 𝐽)
73, 4, 6syl2anc 696 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑁 𝐽)
85ntrss2 21084 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 𝐽) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
93, 7, 8syl2anc 696 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
10 ssrin 3982 . . 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 21076 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 𝐽) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
133, 7, 12syl2anc 696 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
145fclselbas 22042 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
151, 14syl 17 . . . . . . 7 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝐴 𝐽)
1615snssd 4486 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → {𝐴} ⊆ 𝐽)
175neiint 21131 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
183, 16, 7, 17syl3anc 1477 . . . . 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 4460 . . . . 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 1133 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → 𝑆𝐹)
24 fclsopni 22041 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
251, 13, 22, 23, 24syl13anc 1479 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
26 ssn0 4120 . 2 (((((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁𝑆) ∧ (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅) → (𝑁𝑆) ≠ ∅)
2711, 25, 26syl2anc 696 1 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072  wcel 2140  wne 2933  cin 3715  wss 3716  c0 4059  {csn 4322   cuni 4589  cfv 6050  (class class class)co 6815  Topctop 20921  intcnt 21044  neicnei 21124   fClus cfcls 21962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-fbas 19966  df-top 20922  df-topon 20939  df-cld 21046  df-ntr 21047  df-cls 21048  df-nei 21125  df-fil 21872  df-fcls 21967
This theorem is referenced by:  fclsnei  22045  fclsfnflim  22053
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