Theorem iocleub

Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	iocleub	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A (,] B ) ) -> C <_ B )

Proof

Step	Hyp	Ref	Expression
1		elioc1	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
2		simp3	|- ( ( C e. RR* /\ A < C /\ C <_ B ) -> C <_ B )
3	1 2	biimtrdi	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) -> C <_ B ) )
4	3	3impia	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A (,] B ) ) -> C <_ B )