# Ordinal is Transitive

## Theorem

Every ordinal (by Definition 3) is a transitive set.

## Proof

Let $S$ be an ordinal by Definition 3.

By definition:

- $\forall a \in S: a = S_a \subseteq S$

where $S_a$ denotes the initial segment of $S$ determined by $a$.

That is, $S$ is a transitive set.

$\blacksquare$