Minimally Closed Class under Progressing Mapping is Well-Ordered/Proof

Proof
According to hypothesis, let $M$ be a minimally closed under $g$ with respect to $b$.

By Minimally Closed Class under Progressing Mapping induces Nest:
 * $\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$

By Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element:
 * if $x$ is a fixed point of $g$, then $x$ is the greatest element of $M$.

Thus the conditions for Closed Class under Progressing Mapping Lemma hold.

In order to show that $M$ is well-ordered under the inclusion relation, it is to be shown that every non-empty subclass $A$ of $M$ has a smallest element.

Let $A$ be an arbitrary non-empty subclass of $N$.

Let $L$ be the class of all elements $y$ of $N$ such that $y$ is a proper subset of all elements of $A$.

If $b \in A$, then from Smallest Element of Minimally Closed Class under Progressing Mapping, $b$ is the smallest element of $A$.

So, consider the case where $b \notin A$.

Then $b$ is a proper subset of every element of $A$.

Thus $b \in L$ and so $L$ is non-empty.

By definition, $L$ is bounded by all elements of $A$.

Since $A$ is non-empty, $L$ is bounded by at least one element of $A$.

Hence by Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element:
 * $L$ has a greatest element $x$.

Therefore, by Closed Class under Progressing Mapping Lemma, $\map g x$ is the smallest element of $A$.