Metamath Proof Explorer

Theorem elfzel2

Description: Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfzel2 
|- ( K e. ( M ... N ) -> N e. ZZ )

Proof

Step Hyp Ref Expression
1 elfzuz3 
 |-  ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
2 eluzelz 
 |-  ( N e. ( ZZ>= ` K ) -> N e. ZZ )
3 1 2 syl 
 |-  ( K e. ( M ... N ) -> N e. ZZ )