Theorem pnrmnrm 23282

Description: A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmnrm	⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Proof of Theorem pnrmnrm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispnrm 23281	. 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑥)))
2	1	simplbi 497	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Syntax hints:   → wi 4   ∈ wcel 2113   ⊆ wss 3899  ∩ cint 4900   ↦ cmpt 5177  ran crn 5623  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  ℕcn 12143  Clsdccld 22958  Nrmcnrm 23252  PNrmcpnrm 23254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-cnv 5630  df-dm 5632  df-rn 5633  df-iota 6446  df-fv 6498  df-ov 7359  df-pnrm 23261

This theorem is referenced by:  pnrmtop  23283