# Metamath Proof Explorer

## 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 ) )`
` |-  ( ( 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 ) )`