Metamath Proof Explorer

Theorem eluzfz

Description: Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion eluzfz 
|- ( ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> K e. ( M ... N ) )

Proof

Step Hyp Ref Expression
1 elfzuzb 
 |-  ( K e. ( M ... N ) <-> ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) )
2 1 biimpri 
 |-  ( ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> K e. ( M ... N ) )