Theorem ubioc1

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 . (Contributed by FL, 29-May-2014)

	Ref	Expression
Assertion	ubioc1	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
2		simp3	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
3		xrleid	|- ( B e. RR* -> B <_ B )
4	3	3ad2ant2	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B <_ B )
5		elioc1	|- ( ( A e. RR* /\ B e. RR* ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
6	5	3adant3	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
7	1 2 4 6	mpbir3and	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )