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Metamath Proof Explorer

Theorem fzss1

Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005) (Proof shortened by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion fzss1 
|- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )

Proof

Step Hyp Ref Expression
1 elfzuz 
 |-  ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
2 id 
 |-  ( K e. ( ZZ>= ` M ) -> K e. ( ZZ>= ` M ) )
3 uztrn 
 |-  ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
4 1 2 3 syl2anr 
 |-  ( ( K e. ( ZZ>= ` M ) /\ k e. ( K ... N ) ) -> k e. ( ZZ>= ` M ) )
5 elfzuz3 
 |-  ( k e. ( K ... N ) -> N e. ( ZZ>= ` k ) )
6 5 adantl 
 |-  ( ( K e. ( ZZ>= ` M ) /\ k e. ( K ... N ) ) -> N e. ( ZZ>= ` k ) )
7 elfzuzb 
 |-  ( k e. ( M ... N ) <-> ( k e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` k ) ) )
8 4 6 7 sylanbrc 
 |-  ( ( K e. ( ZZ>= ` M ) /\ k e. ( K ... N ) ) -> k e. ( M ... N ) )
9 8 ex 
 |-  ( K e. ( ZZ>= ` M ) -> ( k e. ( K ... N ) -> k e. ( M ... N ) ) )
10 9 ssrdv 
 |-  ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )