Theorem cnrmnrm

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

	Ref	Expression
Assertion	cnrmnrm	$$⊢J\in \mathrm{CNrm}\to J\in \mathrm{Nrm}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\bigcup J=\bigcup J$$
2	1	restid	$$⊢J\in \mathrm{CNrm}\to J{\upharpoonright }_{𝑡}\bigcup J=J$$
3		uniexg	$$⊢J\in \mathrm{CNrm}\to \bigcup J\in \mathrm{V}$$
4		cnrmi	$$⊢\left(J\in \mathrm{CNrm}\wedge \bigcup J\in \mathrm{V}\right)\to J{\upharpoonright }_{𝑡}\bigcup J\in \mathrm{Nrm}$$
5	3 4	mpdan	$$⊢J\in \mathrm{CNrm}\to J{\upharpoonright }_{𝑡}\bigcup J\in \mathrm{Nrm}$$
6	2 5	eqeltrrd	$$⊢J\in \mathrm{CNrm}\to J\in \mathrm{Nrm}$$