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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 ) )
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 ) )