Successor Mapping is Progressing

Theorem
Let $V$ be a basic universe.

Let $s: V \to V$ denote the successor mapping on $V$:
 * $\forall x \in V: \map s x := x \cup \set x$

Then $s$ is a progressing mapping.

Proof
Recall

By Set is Subset of Union:
 * $x \subseteq x \cup \set x$

That is:
 * $x \subseteq \map s x$

Thus $s$ is by definition a progressing mapping.

Proof

 * : Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications