Metamath Proof Explorer

Theorem uzssz

Description: An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005) (Revised by Mario Carneiro, 3-Nov-2013)

Ref Expression
Assertion uzssz 
|- ( ZZ>= ` M ) C_ ZZ

Proof

Step Hyp Ref Expression
1 uzf 
 |-  ZZ>= : ZZ --> ~P ZZ
2 1 ffvelrni 
 |-  ( M e. ZZ -> ( ZZ>= ` M ) e. ~P ZZ )
3 2 elpwid 
 |-  ( M e. ZZ -> ( ZZ>= ` M ) C_ ZZ )
4 1 fdmi 
 |-  dom ZZ>= = ZZ
5 3 4 eleq2s 
 |-  ( M e. dom ZZ>= -> ( ZZ>= ` M ) C_ ZZ )
6 ndmfv 
 |-  ( -. M e. dom ZZ>= -> ( ZZ>= ` M ) = (/) )
7 0ss 
 |-  (/) C_ ZZ
8 6 7 eqsstrdi 
 |-  ( -. M e. dom ZZ>= -> ( ZZ>= ` M ) C_ ZZ )
9 5 8 pm2.61i 
 |-  ( ZZ>= ` M ) C_ ZZ