Intersection of Two Ordinals is Ordinal

Theorem

Let $S, T$ be ordinals.

Then $S \cap T$ is an ordinal.

Proof

Let $a \in S \cap T$.

Then the initial segments $S_a$ and $T_a$ are such that $S_a = a = T_a$.

That is:

$\left\{{x \in S: x \subset a}\right\} = a = \left\{{y \in T: y \subset a}\right\}$.

So:

$a = \left\{{z \in S \cap T: z \subset a}\right\} = \left({S \cap T}\right)_a$.

$\blacksquare$