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Metamath Proof Explorer

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 ) )