No Infinitely Descending Membership Chains

Theorem
Let $\omega$ denote the minimally inductive set.

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

Then:
 * $\exists n \in \omega: \map F {n^+} \notin \map F n$

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

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

Let the image of $F$ be denoted $\map \WW F$.

Then:

But:
 * $\map F {n^+} \in \map \WW F$

So:
 * $\map F {n^+} \notin \map F n$