Union of Slowly Well-Ordered Class under Subset Relation is Well-Orderable/Corollary 3

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

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

In the above, $\bigcup N$ means the union of $N$.

We have that:
 * $\forall a \in \bigcup N:$ if $a$ is not the greatest element of $\bigcup N$, then the immediate successor of $a$ is the smallest element of $\bigcup N \setminus \map F a$.