Theorem pnrmnrm 20954

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 20953	. 2 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑥)))
2	1	simplbi 475	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540  ∩ cint 4410   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  ℕcn 10897  Clsdccld 20630  Nrmcnrm 20924  PNrmcpnrm 20926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-pnrm 20933

This theorem is referenced by:  pnrmtop  20955