Theorem pnrmnrm 23691

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 23690	. 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 2147   ⊆ wss 3900  ∩ cint 4910   ↦ cmpt 5190  ran crn 5657  ‘cfv 6561  (class class class)co 7446   ↑_m cmap 8878  ℕcn 12365  Clsdccld 23367  Nrmcnrm 23661  PNrmcpnrm 23663

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-cnv 5664  df-dm 5666  df-rn 5667  df-iota 6515  df-fv 6569  df-ov 7449  df-pnrm 23670

This theorem is used by:  pnrmtop  23692