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 $\bigcup N \setminus \map F a$.

Proof

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 4$ Well ordering and choice: Exercise $4.1 \ \text{(c)}$