Union of Slowly Well-Ordered Class under Subset Relation is Well-Orderable/Corollary 4

Corollary to Union of Slowly Well-Ordered Class under Subset Relation is Well-Orderable
Let $N$ be a class.

Let $N$ be slowly well-ordered under the subset relation.

For $a \in \ds \bigcup N$, let $\map F a$ denote the smallest element of $N$ that contains $a$.

For $a, b \in \ds \bigcup N$, we define $a \preccurlyeq b \iff \map F a \subseteq \map F b$.

We have that:
 * Every immediate successor element of $N$ is $\map F a$ for some $a \in N$.