Intersection of Two Ordinals is Ordinal

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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:

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

So:

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

$\blacksquare$


Sources