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Theorem locfinnei 26373
 Description: A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
locfinnei.1
Assertion
Ref Expression
locfinnei
Distinct variable groups:   ,,   ,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem locfinnei
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 locfinnei.1 . . . 4
2 eqid 2435 . . . 4
31, 2islocfin 26367 . . 3
43simp3bi 974 . 2
5 eleq1 2495 . . . . 5
65anbi1d 686 . . . 4
76rexbidv 2718 . . 3
87rspccva 3043 . 2
94, 8sylan 458 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698  crab 2701   cin 3311  c0 3620  cuni 4007  cfv 5446  cfn 7101  ctop 16950  clocfin 26333 This theorem is referenced by:  lfinpfin  26374  locfincmp  26375  locfincf  26377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-top 16955  df-locfin 26337
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