Theorem cnrmtop 23258

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 2731	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscnrm 23244	. 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3047  𝒫 cpw 4549  ∪ cuni 4858  (class class class)co 7352   ↾_t crest 17330  Topctop 22814  Nrmcnrm 23231  CNrmccnrm 23232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6443  df-fv 6495  df-ov 7355  df-cnrm 23239

This theorem is referenced by:  restcnrm  23283