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Theorem fclselbas 17807
Description: A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
fclselbas.1  |-  X  = 
U. J
Assertion
Ref Expression
fclselbas  |-  ( A  e.  ( J  fClus  F )  ->  A  e.  X )

Proof of Theorem fclselbas
Dummy variables  o 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fclselbas.1 . . . . . 6  |-  X  = 
U. J
21fclsfil 17801 . . . . 5  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  X ) )
3 fclstopon 17803 . . . . 5  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
42, 3mpbird 223 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  (TopOn `  X ) )
5 fclsopn 17805 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
64, 2, 5syl2anc 642 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  ( A  e.  ( J  fClus  F )  <-> 
( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) ) ) )
76ibi 232 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) )
87simpld 445 1  |-  ( A  e.  ( J  fClus  F )  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619    i^i cin 3227   (/)c0 3531   U.cuni 3906   ` cfv 5334  (class class class)co 5942  TopOnctopon 16732   Filcfil 17636    fClus cfcls 17727
This theorem is referenced by:  fclsneii  17808  fclsnei  17810  fclsfnflim  17818  flimfnfcls  17819  fcfelbas  17827  cnfcf  17833  cfilfcls  18798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-fbas 16473  df-top 16736  df-topon 16739  df-cld 16856  df-ntr 16857  df-cls 16858  df-fil 17637  df-fcls 17732
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