Theorem eliccd

Description: Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	eliccd.1	|- ( ph -> A e. RR )
	eliccd.2	|- ( ph -> B e. RR )
	eliccd.3	|- ( ph -> C e. RR )
	eliccd.4	|- ( ph -> A <_ C )
	eliccd.5	|- ( ph -> C <_ B )
Assertion	eliccd	|- ( ph -> C e. ( A [,] B ) )

Proof

Step	Hyp	Ref	Expression
1		eliccd.1	|- ( ph -> A e. RR )
2		eliccd.2	|- ( ph -> B e. RR )
3		eliccd.3	|- ( ph -> C e. RR )
4		eliccd.4	|- ( ph -> A <_ C )
5		eliccd.5	|- ( ph -> C <_ B )
6		elicc2	|- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A [,] B ) <-> ( C e. RR /\ A <_ C /\ C <_ B ) ) )
7	1 2 6	syl2anc	|- ( ph -> ( C e. ( A [,] B ) <-> ( C e. RR /\ A <_ C /\ C <_ B ) ) )
8	3 4 5 7	mpbir3and	|- ( ph -> C e. ( A [,] B ) )