Metamath Proof Explorer


Theorem elfzuz2

Description: Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfzuz2
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) )

Proof

Step Hyp Ref Expression
1 elfzuzb
 |-  ( K e. ( M ... N ) <-> ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) )
2 eqid
 |-  ( ZZ>= ` M ) = ( ZZ>= ` M )
3 2 uztrn2
 |-  ( ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> N e. ( ZZ>= ` M ) )
4 1 3 sylbi
 |-  ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) )