Theorem ubicc2

Description: The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007) (Revised by FL, 29-May-2014)

	Ref	Expression
Assertion	ubicc2	|- ( ( 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		elicc1	|- ( ( 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 ) )