Theorem cnrmtop 23635

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 2761	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscnrm 23621	. 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
3	2	simplbi 502	1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3077  𝒫 cpw 4557  ∪ cuni 4867  (class class class)co 7412   ↾_t crest 17571  Topctop 23191  Nrmcnrm 23608  CNrmccnrm 23609

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-cnrm 23616

This theorem is used by:  restcnrm  23660