Theorem pnrmnrm 23638

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 23637	. 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 3899  ∩ cint 4907   ↦ cmpt 5186  ran crn 5652  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  ℕcn 12316  Clsdccld 23314  Nrmcnrm 23608  PNrmcpnrm 23610

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-cnv 5659  df-dm 5661  df-rn 5662  df-iota 6487  df-fv 6539  df-ov 7415  df-pnrm 23617

This theorem is used by:  pnrmtop  23639