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Metamath Proof Explorer

Theorem iiconn

Description: The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015)

		Ref	Expression
	Assertion	iiconn	$⊢ II \in Conn$

Step	Hyp	Ref	Expression
1		dfii2	$⊢ II = topGen (ran (.)) ↾_{𝑡} [0, 1]$
2		0re	$⊢ 0 \in ℝ$
3		1re	$⊢ 1 \in ℝ$
4		iccconn	$⊢ (0 \in ℝ \land 1 \in ℝ) \to topGen (ran (.)) ↾_{𝑡} [0, 1] \in Conn$
5	2 3 4	mp2an	$⊢ topGen (ran (.)) ↾_{𝑡} [0, 1] \in Conn$
6	1 5	eqeltri	$⊢ II \in Conn$