Theorem iooltubd

Description: An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

Step	Hyp	Ref	Expression
1		iooltubd.1	|- ( ph -> A e. RR* )
2		iooltubd.2	|- ( ph -> B e. RR* )
3		iooltubd.3	|- ( ph -> C e. ( A (,) B ) )
4		iooltub	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A (,) B ) ) -> C < B )
5	1 2 3 4	syl3anc	|- ( ph -> C < B )