Metamath Proof Explorer


Theorem fznn0

Description: Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005)

Ref Expression
Assertion fznn0 N 0 K 0 N K 0 K N

Proof

Step Hyp Ref Expression
1 0z 0
2 nn0z N 0 N
3 elfz1 0 N K 0 N K 0 K K N
4 1 2 3 sylancr N 0 K 0 N K 0 K K N
5 df-3an K 0 K K N K 0 K K N
6 elnn0z K 0 K 0 K
7 6 anbi1i K 0 K N K 0 K K N
8 5 7 bitr4i K 0 K K N K 0 K N
9 4 8 bitrdi N 0 K 0 N K 0 K N