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

Theorem cnrmtop 17393
Description: A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmtop  |-  ( J  e. CNrm  ->  J  e.  Top )

Proof of Theorem cnrmtop
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3  |-  U. J  =  U. J
21iscnrm 17379 . 2  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
32simplbi 447 1  |-  ( J  e. CNrm  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2697   ~Pcpw 3791   U.cuni 4007  (class class class)co 6073   ↾t crest 13640   Topctop 16950   Nrmcnrm 17366  CNrmccnrm 17367
This theorem is referenced by:  restcnrm  17418
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-pw 3793  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-cnrm 17374
  Copyright terms: Public domain W3C validator