No Infinitely Descending Membership Chains

Theorem
Let $\omega$ denote the minimal infinite successor set.

Let $F$ be a function whose domain is $\omega$.

Then:
 * $\exists n \in \omega: F \left({n^+}\right) \notin F \left({n}\right)$

Proof
Let $F$ be a function whose domain is $\omega$.

By the axiom of replacement, the range of $F$ is a set.

Let the range of $F$ be denoted $\mathcal W \left({F}\right)$.

Then:

But:
 * $F \left({n^+}\right) \in \mathcal W \left({F}\right)$

So:
 * $F \left({n^+}\right) \not \in F \left({n}\right)$