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 ) )