Metamath Proof Explorer

Theorem suprleubii

Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005) (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	suprleubii	\|- ( B e. RR -> ( sup ( A , RR , < ) <_ B <-> A. z e. A z <_ B ) )

Proof

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

Step

Hyp

Ref

Expression

sup3i.1

 |-  ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x )

suprleub

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

1 2

mpan

 |-  ( B e. RR -> ( sup ( A , RR , < ) <_ B <-> A. z e. A z <_ B ) )