Ordinal is Transitive/Proof 3

Theorem

Every ordinal is a transitive set.

Proof

Let $\alpha$ be an ordinal by Definition 3.

An ordinal is a strictly well-ordered set $\struct {\alpha, \prec}$ such that:

$\forall \beta \in \alpha: \alpha_\beta = \beta$

where $\alpha_\beta$ is the initial segment of $\alpha$ determined by $\beta$:

$\alpha_\beta = \set {x \in \alpha: x \prec \beta}$

That is, $\alpha$ is a transitive set.

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$\blacksquare$