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

Theorem cnrmtop 21081
Description: A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmtop (𝐽 ∈ CNrm → 𝐽 ∈ Top)

Proof of Theorem cnrmtop
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 𝐽 = 𝐽
21iscnrm 21067 . 2 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
32simplbi 476 1 (𝐽 ∈ CNrm → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wral 2908  𝒫 cpw 4136   cuni 4409  (class class class)co 6615  t crest 16021  Topctop 20638  Nrmcnrm 21054  CNrmccnrm 21055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-cnrm 21062
This theorem is referenced by:  restcnrm  21106
  Copyright terms: Public domain W3C validator