Theorem pnrmnrm 23571

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 23570	. 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑥)))
2	1	simplbi 502	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3902  ∩ cint 4910   ↦ cmpt 5190  ran crn 5660  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830  ℕcn 12261  Clsdccld 23247  Nrmcnrm 23541  PNrmcpnrm 23543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6493  df-fv 6545  df-ov 7420  df-pnrm 23550

This theorem is used by:  pnrmtop  23572