Theorem conntop

Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008) (Revised by Mario Carneiro, 14-Dec-2013)

	Ref	Expression
Assertion	conntop	⊢ ( 𝐽 ∈ Conn → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn	⊢ ( 𝐽 ∈ Conn ↔ ( 𝐽 ∈ Top ∧ ( 𝐽 ∩ ( Clsd ‘ 𝐽 ) ) = { ∅ , ∪ 𝐽 } ) )
3	2	simplbi	⊢ ( 𝐽 ∈ Conn → 𝐽 ∈ Top )