Theorem fzss2

Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	fzss2	|- ( N e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... N ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuz	|- ( k e. ( M ... K ) -> k e. ( ZZ>= ` M ) )
2	1	adantl	|- ( ( N e. ( ZZ>= ` K ) /\ k e. ( M ... K ) ) -> k e. ( ZZ>= ` M ) )
3		elfzuz3	|- ( k e. ( M ... K ) -> K e. ( ZZ>= ` k ) )
4		uztrn	|- ( ( N e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` k ) ) -> N e. ( ZZ>= ` k ) )
5	3 4	sylan2	|- ( ( N e. ( ZZ>= ` K ) /\ k e. ( M ... K ) ) -> N e. ( ZZ>= ` k ) )
6		elfzuzb	|- ( k e. ( M ... N ) <-> ( k e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` k ) ) )
7	2 5 6	sylanbrc	|- ( ( N e. ( ZZ>= ` K ) /\ k e. ( M ... K ) ) -> k e. ( M ... N ) )
8	7	ex	|- ( N e. ( ZZ>= ` K ) -> ( k e. ( M ... K ) -> k e. ( M ... N ) ) )
9	8	ssrdv	|- ( N e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... N ) )