Class such that Every Transitive Subset is Element of it Contains All Ordinals

Theorem

Let $K$ be a class.

Let $K$ be such that every transitive subset of $K$ is an element of $K$.

Then every ordinal is an element of $K$.

Proof

Let us assume the hypothesis.

Let $\On$ denote the class of all ordinals.

From Well-Ordering of Class of All Ordinals under Subset Relation, $\On$ is well-ordered under $\subseteq$.

Hence we can use the First Principle of Transfinite Induction.

Let $\alpha$ be an ordinal such that every ordinal less than $\alpha$ is an element of $K$.

Thus $\alpha \subseteq K$.

As Ordinal is Transitive, it follows by hypothesis that:

$\alpha \in K$

So if all ordinals less than $\alpha$ are elements of $K$, then so is $\alpha$.

Hence by the First Principle of Transfinite Induction, all ordinals are in $K$.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Theorem $2.4$