Initial Segment of Natural Numbers determined by Zero is Empty

Theorem
Let $\N_k$ denote the initial segment of the natural numbers determined by $k$:
 * $\N_k = \left\{{0, 1, 2, 3, \ldots, k-1}\right\}$

Then $\N_0 = \varnothing$.

Proof
From the definition of $\N_0$:


 * $\N_0 = \left\{{n \in \N: n < 0}\right\}$

From the definition of zero, $0$ is the minimal element of $\N$.

So there is no element $n$ of $\N$ such that $n < 0$.

Thus $\N_0 = \varnothing$.