Theorem cnrmtop 23293

Description: A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
cnrmtop	⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)

Proof of Theorem cnrmtop

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscnrm 23279	. 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052  𝒫 cpw 4556  ∪ cuni 4865  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  Nrmcnrm 23266  CNrmccnrm 23267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-cnrm 23274

This theorem is referenced by:  restcnrm  23318