Theorem eliccelicod

Description: A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	eliccelicod.a	|- ( ph -> A e. RR* )
	eliccelicod.b	|- ( ph -> B e. RR* )
	eliccelicod.c	|- ( ph -> C e. ( A [,] B ) )
	eliccelicod.d	|- ( ph -> C =/= B )
Assertion	eliccelicod	|- ( ph -> C e. ( A [,) B ) )

Proof

Step	Hyp	Ref	Expression
1		eliccelicod.a	|- ( ph -> A e. RR* )
2		eliccelicod.b	|- ( ph -> B e. RR* )
3		eliccelicod.c	|- ( ph -> C e. ( A [,] B ) )
4		eliccelicod.d	|- ( ph -> C =/= B )
5		eliccxr	|- ( C e. ( A [,] B ) -> C e. RR* )
6	3 5	syl	|- ( ph -> C e. RR* )
7		iccgelb	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C )
8	1 2 3 7	syl3anc	|- ( ph -> A <_ C )
9		iccleub	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> C <_ B )
10	1 2 3 9	syl3anc	|- ( ph -> C <_ B )
11	6 2 10 4	xrleneltd	|- ( ph -> C < B )
12	1 2 6 8 11	elicod	|- ( ph -> C e. ( A [,) B ) )