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Theorem fclsneii 18051
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 958 . . . . 5  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  A  e.  ( J  fClus  F ) )
2 fclstop 18045 . . . . 5  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )
31, 2syl 16 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  J  e.  Top )
4 simp2 959 . . . . 5  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  N  e.  ( ( nei `  J
) `  { A } ) )
5 eqid 2438 . . . . . 6  |-  U. J  =  U. J
65neii1 17172 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  N  C_  U. J
)
73, 4, 6syl2anc 644 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  N  C_ 
U. J )
85ntrss2 17123 . . . 4  |-  ( ( J  e.  Top  /\  N  C_  U. J )  ->  ( ( int `  J ) `  N
)  C_  N )
93, 7, 8syl2anc 644 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( int `  J
) `  N )  C_  N )
10 ssrin 3568 . . 3  |-  ( ( ( int `  J
) `  N )  C_  N  ->  ( (
( int `  J
) `  N )  i^i  S )  C_  ( N  i^i  S ) )
119, 10syl 16 . 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 17115 . . . 4  |-  ( ( J  e.  Top  /\  N  C_  U. J )  ->  ( ( int `  J ) `  N
)  e.  J )
133, 7, 12syl2anc 644 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( int `  J
) `  N )  e.  J )
145fclselbas 18050 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  A  e.  U. J )
151, 14syl 16 . . . . . . 7  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  A  e.  U. J )
1615snssd 3945 . . . . . 6  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  { A }  C_  U. J )
175neiint 17170 . . . . . 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 1185 . . . . 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 203 . . . 4  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  { A }  C_  ( ( int `  J ) `  N
) )
20 snssg 3934 . . . . 5  |-  ( A  e.  U. J  -> 
( A  e.  ( ( int `  J
) `  N )  <->  { A }  C_  (
( int `  J
) `  N )
) )
2115, 20syl 16 . . . 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 225 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  A  e.  ( ( int `  J
) `  N )
)
23 simp3 960 . . 3  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  S  e.  F )
24 fclsopni 18049 . . 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 1187 . 2  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  (
( ( int `  J
) `  N )  i^i  S )  =/=  (/) )
26 ssn0 3662 . 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 644 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 178    /\ w3a 937    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   U.cuni 4017   ` cfv 5456  (class class class)co 6083   Topctop 16960   intcnt 17083   neicnei 17163    fClus cfcls 17970
This theorem is referenced by:  fclsnei  18053  fclsfnflim  18061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-fbas 16701  df-top 16965  df-topon 16968  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-fil 17880  df-fcls 17975
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