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Theorem fclsneii 23741
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 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
Proof of Theorem fclsneii
StepHypRef Expression
1 simp1 1136 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝐴 ∈ (𝐽 fClus 𝐹))
2 fclstop 23735 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐽 ∈ Top)
31, 2syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝐽 ∈ Top)
4 simp2 1137 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}))
5 eqid 2732 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
65neii1 22830 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑁 βŠ† βˆͺ 𝐽)
73, 4, 6syl2anc 584 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝑁 βŠ† βˆͺ 𝐽)
85ntrss2 22781 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)
93, 7, 8syl2anc 584 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)
109ssrind 4235 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ (((intβ€˜π½)β€˜π‘) ∩ 𝑆) βŠ† (𝑁 ∩ 𝑆))
115ntropn 22773 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘) ∈ 𝐽)
123, 7, 11syl2anc 584 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ ((intβ€˜π½)β€˜π‘) ∈ 𝐽)
135fclselbas 23740 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
141, 13syl 17 . . . . . . 7 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
1514snssd 4812 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ {𝐴} βŠ† βˆͺ 𝐽)
165neiint 22828 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽 ∧ 𝑁 βŠ† βˆͺ 𝐽) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘)))
173, 15, 7, 16syl3anc 1371 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘)))
184, 17mpbid 231 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘))
19 snssg 4787 . . . . 5 (𝐴 ∈ βˆͺ 𝐽 β†’ (𝐴 ∈ ((intβ€˜π½)β€˜π‘) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘)))
2014, 19syl 17 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ (𝐴 ∈ ((intβ€˜π½)β€˜π‘) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘)))
2118, 20mpbird 256 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝐴 ∈ ((intβ€˜π½)β€˜π‘))
22 simp3 1138 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ 𝑆 ∈ 𝐹)
23 fclsopni 23739 . . 3 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((intβ€˜π½)β€˜π‘) ∈ 𝐽 ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘) ∧ 𝑆 ∈ 𝐹)) β†’ (((intβ€˜π½)β€˜π‘) ∩ 𝑆) β‰  βˆ…)
241, 12, 21, 22, 23syl13anc 1372 . 2 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ (((intβ€˜π½)β€˜π‘) ∩ 𝑆) β‰  βˆ…)
25 ssn0 4400 . 2 (((((intβ€˜π½)β€˜π‘) ∩ 𝑆) βŠ† (𝑁 ∩ 𝑆) ∧ (((intβ€˜π½)β€˜π‘) ∩ 𝑆) β‰  βˆ…) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
2610, 24, 25syl2anc 584 1 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ 𝑆 ∈ 𝐹) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1087   ∈ wcel 2106   β‰  wne 2940   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  Topctop 22615  intcnt 22741  neicnei 22821   fClus cfcls 23660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-topon 22633  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-fil 23570  df-fcls 23665
This theorem is referenced by:  fclsnei  23743  fclsfnflim  23751
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