Theorem pconntop

Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Assertion	pconntop	⊢ ( 𝐽 ∈ PConn → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ispconn	⊢ ( 𝐽 ∈ PConn ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ∃ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ) )
3	2	simplbi	⊢ ( 𝐽 ∈ PConn → 𝐽 ∈ Top )