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Metamath Proof Explorer

Theorem fzofi

Description: Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015)

Ref Expression
Assertion fzofi 
|- ( M ..^ N ) e. Fin

Proof

Step Hyp Ref Expression
1 fzoval 
 |-  ( N e. ZZ -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
2 1 adantl 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
3 fzfi 
 |-  ( M ... ( N - 1 ) ) e. Fin
4 2 3 eqeltrdi 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M ..^ N ) e. Fin )
5 fzof 
 |-  ..^ : ( ZZ X. ZZ ) --> ~P ZZ
6 5 fdmi 
 |-  dom ..^ = ( ZZ X. ZZ )
7 6 ndmov 
 |-  ( -. ( M e. ZZ /\ N e. ZZ ) -> ( M ..^ N ) = (/) )
8 0fi 
 |-  (/) e. Fin
9 7 8 eqeltrdi 
 |-  ( -. ( M e. ZZ /\ N e. ZZ ) -> ( M ..^ N ) e. Fin )
10 4 9 pm2.61i 
 |-  ( M ..^ N ) e. Fin