Theorem haustop

Description: A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007)

	Ref	Expression
Assertion	haustop	⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ishaus	⊢ ( 𝐽 ∈ Haus ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( 𝑥 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) )
3	2	simplbi	⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ Top )