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Theorem fclsopni 20641
Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsopni  |-  ( ( A  e.  ( J 
fClus  F )  /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F )
)  ->  ( U  i^i  S )  =/=  (/) )

Proof of Theorem fclsopni
Dummy variables  o 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . . . . . . 9  |-  U. J  =  U. J
21fclsfil 20636 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  U. J
) )
3 fclstopon 20638 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  U. J )  <-> 
F  e.  ( Fil `  U. J ) ) )
42, 3mpbird 232 . . . . . . 7  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  (TopOn `  U. J ) )
5 fclsopn 20640 . . . . . . 7  |-  ( ( J  e.  (TopOn `  U. J )  /\  F  e.  ( Fil `  U. J ) )  -> 
( A  e.  ( J  fClus  F )  <->  ( A  e.  U. J  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) ) ) )
64, 2, 5syl2anc 661 . . . . . 6  |-  ( A  e.  ( J  fClus  F )  ->  ( A  e.  ( J  fClus  F )  <-> 
( A  e.  U. J  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
76ibi 241 . . . . 5  |-  ( A  e.  ( J  fClus  F )  ->  ( A  e.  U. J  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) )
87simprd 463 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) )
9 eleq2 2530 . . . . . 6  |-  ( o  =  U  ->  ( A  e.  o  <->  A  e.  U ) )
10 ineq1 3689 . . . . . . . 8  |-  ( o  =  U  ->  (
o  i^i  s )  =  ( U  i^i  s ) )
1110neeq1d 2734 . . . . . . 7  |-  ( o  =  U  ->  (
( o  i^i  s
)  =/=  (/)  <->  ( U  i^i  s )  =/=  (/) ) )
1211ralbidv 2896 . . . . . 6  |-  ( o  =  U  ->  ( A. s  e.  F  ( o  i^i  s
)  =/=  (/)  <->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) )
139, 12imbi12d 320 . . . . 5  |-  ( o  =  U  ->  (
( A  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) )  <->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) ) )
1413rspccv 3207 . . . 4  |-  ( A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) )  ->  ( U  e.  J  ->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s
)  =/=  (/) ) ) )
158, 14syl 16 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) ) )
16 ineq2 3690 . . . . 5  |-  ( s  =  S  ->  ( U  i^i  s )  =  ( U  i^i  S
) )
1716neeq1d 2734 . . . 4  |-  ( s  =  S  ->  (
( U  i^i  s
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1817rspccv 3207 . . 3  |-  ( A. s  e.  F  ( U  i^i  s )  =/=  (/)  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) )
1915, 18syl8 70 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) ) ) )
20193imp2 1211 1  |-  ( ( A  e.  ( J 
fClus  F )  /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F )
)  ->  ( U  i^i  S )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    i^i cin 3470   (/)c0 3793   U.cuni 4251   ` cfv 5594  (class class class)co 6296  TopOnctopon 19521   Filcfil 20471    fClus cfcls 20562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-fbas 18542  df-top 19525  df-topon 19528  df-cld 19646  df-ntr 19647  df-cls 19648  df-fil 20472  df-fcls 20567
This theorem is referenced by:  fclsneii  20643  supnfcls  20646  flimfnfcls  20654  cfilfcls  21838
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