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Theorem nllytop 20158
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 20154 . 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 458 1  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   A.wral 2753   E.wrex 2754    i^i cin 3412   ~Pcpw 3954   {csn 3971   ` cfv 5525  (class class class)co 6234   ↾t crest 14927   Topctop 19578   neicnei 19783  𝑛Locally cnlly 20150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5489  df-fv 5533  df-ov 6237  df-nlly 20152
This theorem is referenced by:  nlly2i  20161  restnlly  20167  nllyrest  20171  nllyidm  20174  cldllycmp  20180  llycmpkgen  20237  txnlly  20322  txkgen  20337  xkococnlem  20344  xkococn  20345  cnmptkk  20368  xkofvcn  20369  cnmptk1p  20370  cnmptk2  20371  xkocnv  20499  xkohmeo  20500
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