Theorem t0top

Description: A T_0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010)

	Ref	Expression
Assertion	t0top	⊢ ( 𝐽 ∈ Kol2 → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist0	⊢ ( 𝐽 ∈ Kol2 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
3	2	simplbi	⊢ ( 𝐽 ∈ Kol2 → 𝐽 ∈ Top )