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Metamath Proof Explorer

Theorem suprub

Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004)

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

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> A C_ RR )
2 1 sselda 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ B e. A ) -> B e. RR )
3 suprcl 
 |-  ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
4 3 adantr 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ B e. A ) -> sup ( A , RR , < ) e. RR )
5 ltso 
 |-  < Or RR
6 5 a1i 
 |-  ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> < Or RR )
7 sup3 
 |-  ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
8 6 7 supub 
 |-  ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> ( B e. A -> -. sup ( A , RR , < ) < B ) )
9 8 imp 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ B e. A ) -> -. sup ( A , RR , < ) < B )
10 2 4 9 nltled 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ B e. A ) -> B <_ sup ( A , RR , < ) )