Theorem icoltubd

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

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

Proof

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