No Infinitely Descending Membership Chains/Corollary

Theorem
There cannot exist a sequence $\left\langle{x_n}\right\rangle$ whose domain is $\N_{\gt 0}$ such that:
 * $\forall n \in \N_{\gt 0}: x_{n+1} \in x_n$

Proof
Aiming for contradiction, suppose that there is a sequence like that.

From the definition of a sequence, let $f$ be the the mapping that is defined by $\left\langle{x_n}\right\rangle$.

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

Let $g: \omega \to \N_{\gt 0}$ be defined as:
 * $g \left({\alpha}\right) = \alpha + 1$

Then the composition $f \circ g$ is a mapping whose domain is $\omega$ such that:
 * $\forall n \in \omega: \left({f \circ g}\right) \left({n^+}\right) \in \left({f \circ g}\right) \left({n}\right)$

But this contradicts No Infinitely Descending Membership Chains.

Therefore by contradiction there cannot exist such a sequence.

Hence the result.