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Theorem llytop 20471
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 20467 . 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 461 1  |-  ( J  e. Locally  A  ->  J  e. 
Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1868   A.wral 2775   E.wrex 2776    i^i cin 3435   ~Pcpw 3979  (class class class)co 6301   ↾t crest 15304   Topctop 19901  Locally clly 20463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-iota 5561  df-fv 5605  df-ov 6304  df-lly 20465
This theorem is referenced by:  llynlly  20476  islly2  20483  llyrest  20484  llyidm  20487  nllyidm  20488  toplly  20489  lly1stc  20495  txlly  20635
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