Metamath Proof Explorer


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 ) )