Theorem regtop

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

	Ref	Expression
Assertion	regtop	⊢ ( 𝐽 ∈ Reg → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		isreg	⊢ ( 𝐽 ∈ Reg ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝐽 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) )
2	1	simplbi	⊢ ( 𝐽 ∈ Reg → 𝐽 ∈ Top )