Metamath Proof Explorer


Theorem suprlubii

Description: The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004) (Revised by Mario Carneiro, 6-Sep-2014)

Ref Expression
Hypothesis sup3i.1
|- ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x )
Assertion suprlubii
|- ( B e. RR -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )

Proof

Step Hyp Ref Expression
1 sup3i.1
 |-  ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x )
2 suprlub
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ B e. RR ) -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )
3 1 2 mpan
 |-  ( B e. RR -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )