Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element

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

Let $x$ be a fixed point of $g$.

Then $x$ is the greatest element of $M$.

Proof
Let $x$ be an element of $M$ such that $\map g x = x$.

From Empty Set is Subset of All Sets, we have that:
 * $\O \subseteq x$

Suppose that $y \subseteq x$.

Then by Characteristics of Minimally Inductive Class under Progressing Mapping:
 * $\map g y \subseteq \map g x$

But we have that $\map g x = x$.

Thus:
 * $\map g y \subseteq x$

That is:
 * $\O \subseteq x$

and:
 * $y \subseteq x \implies \map g y \subseteq x$

and the result follows by the Principle of General Induction.