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Theorem llytop 17198
Description: A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llytop  |-  ( J  e. Locally  A  ->  J  e. 
Top )

Proof of Theorem llytop
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islly 17194 . 2  |-  ( J  e. Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( J  i^i  ~P x ) ( y  e.  u  /\  ( Jt  u )  e.  A
) ) )
21simplbi 446 1  |-  ( J  e. Locally  A  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   ~Pcpw 3625  (class class class)co 5858   ↾t crest 13325   Topctop 16631  Locally clly 17190
This theorem is referenced by:  llynlly  17203  islly2  17210  llyrest  17211  llyidm  17214  nllyidm  17215  toplly  17216  lly1stc  17222  txlly  17330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lly 17192
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