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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 ) )