Theorem nrmtop

Description: A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010)

	Ref	Expression
Assertion	nrmtop	⊢ ( 𝐽 ∈ Nrm → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		isnrm	⊢ ( 𝐽 ∈ Nrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ ( ( Clsd ‘ 𝐽 ) ∩ 𝒫 𝑥 ) ∃ 𝑧 ∈ 𝐽 ( 𝑦 ⊆ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) )
2	1	simplbi	⊢ ( 𝐽 ∈ Nrm → 𝐽 ∈ Top )