Union of Slowly Well-Ordered Class under Subset Relation is Well-Orderable/Corollary 2

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 $a \preccurlyeq b \iff \map F a \subseteq \map F b$.

We have that:
 * $\forall x \in N, a \in \bigcup N: x \in \map F a \implies x \preccurlyeq a$