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Theorem locfinnei 19852
 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 2467 . . . 4
31, 2islocfin 19846 . . 3
43simp3bi 1013 . 2
5 eleq1 2539 . . . . 5
65anbi1d 704 . . . 4
76rexbidv 2973 . . 3
87rspccva 3213 . 2
94, 8sylan 471 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767   wne 2662  wral 2814  wrex 2815  crab 2818   cin 3475  c0 3785  cuni 4245  cfv 5588  cfn 7517  ctop 19201  clocfin 19812 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-top 19206  df-locfin 19816 This theorem is referenced by:  lfinpfin  19853  locfincmp  19854  locfincf  19859
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