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Theorem nllytop 17528
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 17524 . 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 447 1  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311   ~Pcpw 3791   {csn 3806   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950   neicnei 17153  𝑛Locally cnlly 17520
This theorem is referenced by:  nlly2i  17531  restnlly  17537  nllyrest  17541  nllyidm  17544  cldllycmp  17550  llycmpkgen  17576  txnlly  17661  txkgen  17676  xkococnlem  17683  xkococn  17684  cnmptkk  17707  xkofvcn  17708  cnmptk1p  17709  cnmptk2  17710  xkocnv  17838  xkohmeo  17839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlly 17522
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