Theorem suprnub

Description: An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Nov-2004) (Revised by Mario Carneiro, 6-Sep-2014)

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

Proof

Step	Hyp	Ref	Expression
1		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 ) )
2	1	notbid	|- ( ( ( 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		ralnex	|- ( A. z e. A -. B < z <-> -. E. z e. A B < z )
4	2 3	bitr4di	|- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ B e. RR ) -> ( -. B < sup ( A , RR , < ) <-> A. z e. A -. B < z ) )