Successor Mapping on Natural Numbers is Progressing

Theorem
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.

Let $s: \omega \to \omega$ denote the successor mapping on $\omega$.

Then $s$ is a progressing mapping.

Proof
By definition of the von Neumann construction:


 * $n^+ = n \cup \set n$

from which it follows that:
 * $n \subseteq n^+$