Theorem lbico1

Description: The lower bound belongs to a closed-below, open-above interval. See lbicc2 . (Contributed by FL, 29-May-2014)

	Ref	Expression
Assertion	lbico1	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* )
2		xrleid	|- ( A e. RR* -> A <_ A )
3	2	3ad2ant1	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A )
4		simp3	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
5		elico1	|- ( ( A e. RR* /\ B e. RR* ) -> ( A e. ( A [,) B ) <-> ( A e. RR* /\ A <_ A /\ A < B ) ) )
6	5	3adant3	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( A e. ( A [,) B ) <-> ( A e. RR* /\ A <_ A /\ A < B ) ) )
7	1 3 4 6	mpbir3and	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )