Intersection of Two Ordinals is Ordinal

Theorem
Let $S$ and $T$ be ordinals.

Then $S \cap T$ is an ordinal.

Proof
Because $S$ and $T$ are ordinals, they are (strictly) well-ordered by the subset relation.

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:
 * $\set {x \in S: x \subset a} = a = \set {y \in T: y \subset a}$

So:
 * $a = \set {z \in S \cap T: z \subset a} = \paren {S \cap T}_a$

Hence it is seen that $\paren {S \cap T}_a$ is an initial segment of both $S$ and $T$.

The result follows from Initial Segment of Ordinal is Ordinal.