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Theorem nllytop 17199
Description: A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllytop  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )

Proof of Theorem nllytop
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 17195 . 2  |-  ( J  e. 𝑛Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
21simplbi 446 1  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   neicnei 16834  𝑛Locally cnlly 17191
This theorem is referenced by:  nlly2i  17202  restnlly  17208  nllyrest  17212  nllyidm  17215  cldllycmp  17221  llycmpkgen  17247  txnlly  17331  txkgen  17346  xkococnlem  17353  xkococn  17354  cnmptkk  17377  xkofvcn  17378  cnmptk1p  17379  cnmptk2  17380  xkocnv  17505  xkohmeo  17506
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-nlly 17193
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