Theorem pnrmnrm 22104

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 22103	. 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑥)))
2	1	simplbi 501	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3853  ∩ cint 4846   ↦ cmpt 5120  ran crn 5536  ‘cfv 6350  (class class class)co 7183   ↑_m cmap 8450  ℕcn 11729  Clsdccld 21780  Nrmcnrm 22074  PNrmcpnrm 22076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rab 3063  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-cnv 5543  df-dm 5545  df-rn 5546  df-iota 6308  df-fv 6358  df-ov 7186  df-pnrm 22083

This theorem is referenced by:  pnrmtop  22105