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

Theorem t0top 17057
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 2283 . . 3  |-  U. J  =  U. J
21ist0 17048 . 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 1623    e. wcel 1684   A.wral 2543   U.cuni 3827   Topctop 16631   Kol2ct0 17034
This theorem is referenced by:  restt0  17094  sst0  17101  kqt0  17437  t0hmph  17481  kqhmph  17510  ordtopt0  24881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-uni 3828  df-t0 17041
  Copyright terms: Public domain W3C validator