Theorem cnrmtop 23531

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 2766	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscnrm 23517	. 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 2146  ∀wral 3082  𝒫 cpw 4567  ∪ cuni 4877  (class class class)co 7423   ↾_t crest 17498  Topctop 23087  Nrmcnrm 23504  CNrmccnrm 23505

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 2148  ax-9 2156  ax-ext 2738

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-cnrm 23512

This theorem is used by:  restcnrm  23556