Set of Strictly Negative Integers is Primitive Recursive

Theorem
Let $N \subseteq \N$ be the set of all $n \in \N$ such that:
 * $n$ codes an integer $k$ such that $k < 0$.

Then $N$ is a primitive recursive set.

Proof
By Set of Strictly Positive Integers is Primitive Recursive:
 * $P = \set {n \in \N : k > 0}$

is primitive recursive.

By Complement of Primitive Recursive Set:
 * $P^c = \set {n \in \N : k \le 0}$

is primitive recursive.

It is clear that:
 * $N = P^c \setminus \set {n \in \N : k = 0}$

By Set Difference as Intersection with Relative Complement:
 * $N = P^c \cap \relcomp \N {\set {n \in \N : k = 0}}$

By: we only need to show that:
 * Intersection of Primitive Recursive Sets
 * Complement of Primitive Recursive Set
 * $\set {n \in \N : k = 0}$

is primitive recursive.

As $k = 0 \le 0$:
 * $n = - 2 k = 0$

Therefore:
 * $\set {n \in \N : k = 0} = \set 0$

which is primitive recursive by:
 * Set Containing Only Zero is Primitive Recursive