Theorem iocleubd

Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021)

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

Proof

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