Theorem icoltub

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

	Ref	Expression
Assertion	icoltub	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,) B ) ) -> C < B )

Proof

Step	Hyp	Ref	Expression
1		elico1	|- ( ( 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	syl6bi	|- ( ( 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 )