Minimally Inductive Class under Progressing Mapping with Fixed Element is Finite

Theorem
Let $M$ be a class which is minimally inductive under a progressing mapping $g$.

Let there exist an element $x \in M$ such that $x = \map g x$.

Then $M$ is a finite class.

Proof
By Set of Subsets of Element of Minimally Inductive Class under Progressing Mapping is Finite, the set:
 * $\set {y \in M: y \subseteq x}$

is finite.

From Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element, $x$ is the greatest element of $M$.

Thus:
 * $M = \set {y \in M: y \subseteq x}$

Hence the result.