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

Theorem cmptop 17122
Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
Assertion
Ref Expression
cmptop  |-  ( J  e.  Comp  ->  J  e. 
Top )

Proof of Theorem cmptop
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  U. J  =  U. J
21iscmp 17115 . 2  |-  ( J  e.  Comp  <->  ( J  e. 
Top  /\  A. r  e.  ~P  J ( U. J  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) U. J  =  U. s ) ) )
32simplbi 446 1  |-  ( J  e.  Comp  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   U.cuni 3827   Fincfn 6863   Topctop 16631   Compccmp 17113
This theorem is referenced by:  imacmp  17124  cmpcld  17129  fiuncmp  17131  cmpfii  17136  kgeni  17232  kgentopon  17233  kgencmp  17240  kgencmp2  17241  cmpkgen  17246  txcmplem1  17335  txcmp  17337  qtopcmp  17399  cmphaushmeo  17491  ptcmpfi  17504  fclscmpi  17724  alexsubALTlem1  17741  ptcmplem1  17746  ptcmpg  17751  evth  18457  evth2  18458  ordcmp  24886  bwt2  25592  locfincmp  26304  heibor1lem  26533  cmpfiiin  26772  kelac1  27161  kelac2  27163  stoweidlem28  27777  stoweidlem50  27799  stoweidlem53  27802  stoweidlem57  27806  stoweidlem59  27808  stoweidlem62  27811
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-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-cmp 17114
  Copyright terms: Public domain W3C validator