Theorem pnrmnrm 23327

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 23326	. 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑥)))
2	1	simplbi 498	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Syntax hints:   → wi 4   ∈ wcel 2121   ⊆ wss 3885  ∩ cint 4880   ↦ cmpt 5156  ran crn 5622  ‘cfv 6489  (class class class)co 7360   ↑_m cmap 8767  ℕcn 12169  Clsdccld 23003  Nrmcnrm 23297  PNrmcpnrm 23299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-cnv 5629  df-dm 5631  df-rn 5632  df-iota 6445  df-fv 6497  df-ov 7363  df-pnrm 23306

This theorem is referenced by:  pnrmtop  23328