Metamath Proof Explorer

Theorem elfz4

Description: Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfz4 
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) -> K e. ( M ... N ) )

Proof

Step Hyp Ref Expression
1 elfz2 
 |-  ( K e. ( M ... N ) <-> ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) )
2 1 biimpri 
 |-  ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) -> K e. ( M ... N ) )