Theorem icogelbd

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

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

Proof

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