Theorem ioogtlb

Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		elioo2	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <-> ( C e. RR /\ A < C /\ C < B ) ) )
2		simp2	|- ( ( C e. RR /\ A < C /\ C < B ) -> A < C )
3	1 2	biimtrdi	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) -> A < C ) )
4	3	3impia	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A (,) B ) ) -> A < C )