Theorem lbicc2

Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007) (Revised by FL, 29-May-2014) (Revised by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	lbicc2	|- ( ( 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		elicc1	|- ( ( 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 ) )