Theorem iocgtlbd

Description: An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022)

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

Proof

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