No Infinitely Descending Membership Chains/Corollary

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

Proof
there is a sequence like that.

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

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

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

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

But this contradicts No Infinitely Descending Membership Chains.

Therefore by Proof by Contradiction there cannot exist such a sequence.

Hence the result.