No Infinitely Descending Membership Chains

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

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

Then, $\exists n: F(n^+) \not \prec F(n)$.

Proof
The domain of $F$ is $\omega$. By the axiom of replacement, the range of $F$ is a set, and the function itself is a set. We shall denote the range of $F$ by $\mathscr{W} (F)$.

But $F(n^+) \in \mathscr{W}(F)$, so $F(n^+) \not \in F(n)$.