Metamath Proof Explorer


Theorem iscnrm4

Description: A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024)

Ref Expression
Assertion iscnrm4 J CNrm J Top s 𝒫 J t 𝒫 J s cls J t = cls J s t = n nei J s m nei J t n m =

Proof

Step Hyp Ref Expression
1 iscnrm3 J CNrm J Top s 𝒫 J t 𝒫 J s cls J t = cls J s t = n J m J s n t m n m =
2 id J Top J Top
3 2 sepnsepo J Top n nei J s m nei J t n m = n J m J s n t m n m =
4 3 imbi2d J Top s cls J t = cls J s t = n nei J s m nei J t n m = s cls J t = cls J s t = n J m J s n t m n m =
5 4 2ralbidv J Top s 𝒫 J t 𝒫 J s cls J t = cls J s t = n nei J s m nei J t n m = s 𝒫 J t 𝒫 J s cls J t = cls J s t = n J m J s n t m n m =
6 5 pm5.32i J Top s 𝒫 J t 𝒫 J s cls J t = cls J s t = n nei J s m nei J t n m = J Top s 𝒫 J t 𝒫 J s cls J t = cls J s t = n J m J s n t m n m =
7 1 6 bitr4i J CNrm J Top s 𝒫 J t 𝒫 J s cls J t = cls J s t = n nei J s m nei J t n m =