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