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Theorem cnrmtop 23185
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 2724 . . 3 𝐽 = 𝐽
21iscnrm 23171 . 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 2098  wral 3053  𝒫 cpw 4595   cuni 4900  (class class class)co 7402  t crest 17371  Topctop 22739  Nrmcnrm 23158  CNrmccnrm 23159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-cnrm 23166
This theorem is referenced by:  restcnrm  23210
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