Theorem ioogtlbd

Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

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