Transitive Class of Ordinals is Subset of Ordinal not in it

Theorem

Let $A$ be a transitive class of ordinals.

Let $\alpha$ be an ordinal which is not an element of $A$.

Then:

$A \subseteq \alpha$

Proof

Let $A$ and $\alpha$ be by hypothesis.

Let $\beta \in A$ be arbitrary.

Because $\beta \in A$ and $\alpha \notin A$ we have that:

$\beta \ne \alpha$

Because $\beta \in A$ and $A$ is transitive:

all elements of $\beta$ are in $A$

But because $\alpha \notin A$:

$\alpha \notin \beta$

Thus we have:

$\alpha \notin \beta$

and:

$\alpha \ne \beta$

Hence from Ordinal Membership is Trichotomy:

$\beta \in \alpha$

As $\beta$ is arbitrary, it follows that:

$A \subseteq \alpha$

$\blacksquare$

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: Lemma $2.1$