MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t0top Unicode version

Theorem t0top 17073
Description: A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
t0top  |-  ( J  e.  Kol2  ->  J  e. 
Top )

Proof of Theorem t0top
Dummy variables  x  y  o are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . 3  |-  U. J  =  U. J
21ist0 17064 . 2  |-  ( J  e.  Kol2  <->  ( J  e. 
Top  /\  A. x  e.  U. J A. y  e.  U. J ( A. o  e.  J  (
x  e.  o  <->  y  e.  o )  ->  x  =  y ) ) )
32simplbi 446 1  |-  ( J  e.  Kol2  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   U.cuni 3843   Topctop 16647   Kol2ct0 17050
This theorem is referenced by:  restt0  17110  sst0  17117  kqt0  17453  t0hmph  17497  kqhmph  17526  ordtopt0  24953
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-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-uni 3844  df-t0 17057
  Copyright terms: Public domain W3C validator