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Theorem fclsneii 17712
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  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  ( N  i^i  S )  =/=  (/) )

Proof of Theorem fclsneii
StepHypRef Expression
1 simp1 955 . . . . 5  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  A  e.  ( J  fClus  F ) )
2 fclstop 17706 . . . . 5  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )
31, 2syl 15 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  J  e.  Top )
4 simp2 956 . . . . 5  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  N  e.  ( ( nei `  J
) `  { A } ) )
5 eqid 2283 . . . . . 6  |-  U. J  =  U. J
65neii1 16843 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  N  C_  U. J
)
73, 4, 6syl2anc 642 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  N  C_ 
U. J )
85ntrss2 16794 . . . 4  |-  ( ( J  e.  Top  /\  N  C_  U. J )  ->  ( ( int `  J ) `  N
)  C_  N )
93, 7, 8syl2anc 642 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( int `  J
) `  N )  C_  N )
10 ssrin 3394 . . 3  |-  ( ( ( int `  J
) `  N )  C_  N  ->  ( (
( int `  J
) `  N )  i^i  S )  C_  ( N  i^i  S ) )
119, 10syl 15 . 2  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( ( int `  J
) `  N )  i^i  S )  C_  ( N  i^i  S ) )
125ntropn 16786 . . . 4  |-  ( ( J  e.  Top  /\  N  C_  U. J )  ->  ( ( int `  J ) `  N
)  e.  J )
133, 7, 12syl2anc 642 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( int `  J
) `  N )  e.  J )
145fclselbas 17711 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  A  e.  U. J )
151, 14syl 15 . . . . . . 7  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  A  e.  U. J )
1615snssd 3760 . . . . . 6  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  { A }  C_  U. J )
175neiint 16841 . . . . . 6  |-  ( ( J  e.  Top  /\  { A }  C_  U. J  /\  N  C_  U. J
)  ->  ( N  e.  ( ( nei `  J
) `  { A } )  <->  { A }  C_  ( ( int `  J ) `  N
) ) )
183, 16, 7, 17syl3anc 1182 . . . . 5  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  ( N  e.  ( ( nei `  J ) `  { A } )  <->  { A }  C_  ( ( int `  J ) `  N
) ) )
194, 18mpbid 201 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  { A }  C_  ( ( int `  J ) `  N
) )
20 snssg 3754 . . . . 5  |-  ( A  e.  U. J  -> 
( A  e.  ( ( int `  J
) `  N )  <->  { A }  C_  (
( int `  J
) `  N )
) )
2115, 20syl 15 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  ( A  e.  ( ( int `  J ) `  N )  <->  { A }  C_  ( ( int `  J ) `  N
) ) )
2219, 21mpbird 223 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  A  e.  ( ( int `  J
) `  N )
)
23 simp3 957 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  S  e.  F )
24 fclsopni 17710 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  (
( ( int `  J
) `  N )  e.  J  /\  A  e.  ( ( int `  J
) `  N )  /\  S  e.  F
) )  ->  (
( ( int `  J
) `  N )  i^i  S )  =/=  (/) )
251, 13, 22, 23, 24syl13anc 1184 . 2  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( ( int `  J
) `  N )  i^i  S )  =/=  (/) )
26 ssn0 3487 . 2  |-  ( ( ( ( ( int `  J ) `  N
)  i^i  S )  C_  ( N  i^i  S
)  /\  ( (
( int `  J
) `  N )  i^i  S )  =/=  (/) )  -> 
( N  i^i  S
)  =/=  (/) )
2711, 25, 26syl2anc 642 1  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  ( N  i^i  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   Topctop 16631   intcnt 16754   neicnei 16834    fClus cfcls 17631
This theorem is referenced by:  fclsnei  17714  fclsfnflim  17722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-fbas 17520  df-fil 17541  df-fcls 17636
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