Theorem eluzfz1

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

	Ref	Expression
Assertion	eluzfz1	|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )

Proof

Step	Hyp	Ref	Expression
1		eluzel2	|- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
2		uzid	|- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1 2	syl	|- ( N e. ( ZZ>= ` M ) -> M e. ( ZZ>= ` M ) )
4		eluzfz	|- ( ( M e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` M ) ) -> M e. ( M ... N ) )
5	3 4	mpancom	|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )