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Theorem nllytop 17215
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 17211 . 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 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   ~Pcpw 3638   {csn 3653   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647   neicnei 16850  𝑛Locally cnlly 17207
This theorem is referenced by:  nlly2i  17218  restnlly  17224  nllyrest  17228  nllyidm  17231  cldllycmp  17237  llycmpkgen  17263  txnlly  17347  txkgen  17362  xkococnlem  17369  xkococn  17370  cnmptkk  17393  xkofvcn  17394  cnmptk1p  17395  cnmptk2  17396  xkocnv  17521  xkohmeo  17522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-nlly 17209
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