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Theorem cnrmtop 23240
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 2728 . . 3 𝐽 = 𝐽
21iscnrm 23226 . 2 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
32simplbi 497 1 (𝐽 ∈ CNrm → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wral 3058  𝒫 cpw 4603   cuni 4908  (class class class)co 7420  t crest 17401  Topctop 22794  Nrmcnrm 23213  CNrmccnrm 23214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-cnrm 23221
This theorem is referenced by:  restcnrm  23265
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