Theorem iillysconn

Description: The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015)

	Ref	Expression
Assertion	iillysconn	⊢ II ∈ Locally SConn

Proof

Step	Hyp	Ref	Expression
1		dfii2	⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) )
2		0re	⊢ 0 ∈ ℝ
3		1re	⊢ 1 ∈ ℝ
4		iccllysconn	⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) ) ∈ Locally SConn )
5	2 3 4	mp2an	⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) ) ∈ Locally SConn
6	1 5	eqeltri	⊢ II ∈ Locally SConn