Theorem eliood

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

	Ref	Expression
Hypotheses	eliood.1	|- ( ph -> A e. RR* )
	eliood.2	|- ( ph -> B e. RR* )
	eliood.3	|- ( ph -> C e. RR )
	eliood.4	|- ( ph -> A < C )
	eliood.5	|- ( ph -> C < B )
Assertion	eliood	|- ( ph -> C e. ( A (,) B ) )

Proof

Step	Hyp	Ref	Expression
1		eliood.1	|- ( ph -> A e. RR* )
2		eliood.2	|- ( ph -> B e. RR* )
3		eliood.3	|- ( ph -> C e. RR )
4		eliood.4	|- ( ph -> A < C )
5		eliood.5	|- ( ph -> C < B )
6		elioo2	|- ( ( 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 ) )