Theorem pnrmnrm 23255

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 23254	. 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 2111   ⊆ wss 3897  ∩ cint 4895   ↦ cmpt 5170  ran crn 5615  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  ℕcn 12125  Clsdccld 22931  Nrmcnrm 23225  PNrmcpnrm 23227

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-cnv 5622  df-dm 5624  df-rn 5625  df-iota 6437  df-fv 6489  df-ov 7349  df-pnrm 23234

This theorem is referenced by:  pnrmtop  23256