G-Tower is Well-Ordered under Subset Relation/Successor of Non-Greatest Element

Theorem
Let $M$ be a class.

Let $g: M \to M$ be a progressing mapping on $M$.

Let $M$ be a $g$-tower.

Let $x \in M$ such that $x$ is not the greatest element of $M$.

Then the immediate successor of $x$ is $\map g x$.

Proof
We have that $g$-Tower is Well-Ordered under Subset Relation.

Let $x \in M$ such that $x$ is not the greatest element of $M$.

Then from Fixed Point of $g$-Tower is Greatest Element:
 * $x \ne \map g x$

Hence:
 * $x \subsetneqq \map g x$

Hence by the Sandwich Principle for $g$-Towers, there is no $y \in M$ such that:
 * $x \subsetneqq y \subsetneqq \map g x$

Hence $\map g x$ is the immediate successor of $x$.