Intersection of Two Ordinals is Ordinal

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Let $S$ and $T$ be ordinals.

Then $S \cap T$ is an ordinal.


Because $S$ and $T$ are ordinals, a fortiori 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:

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


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

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.