MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclsneii Unicode version

Theorem fclsneii 17728
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 17722 . . . . 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 2296 . . . . . 6  |-  U. J  =  U. J
65neii1 16859 . . . . 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 16810 . . . 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 3407 . . 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 16802 . . . 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 17727 . . . . . . . 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 3776 . . . . . 6  |-  ( ( A  e.  ( J 
fClus  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  F )  ->  { A }  C_  U. J )
175neiint 16857 . . . . . 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 3767 . . . . 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 17726 . . 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 3500 . 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 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   ` cfv 5271  (class class class)co 5874   Topctop 16647   intcnt 16770   neicnei 16850    fClus cfcls 17647
This theorem is referenced by:  fclsnei  17730  fclsfnflim  17738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-fbas 17536  df-fil 17557  df-fcls 17652
  Copyright terms: Public domain W3C validator