Ordinal is Transitive

Theorem

Every ordinal is a transitive set.

Proof 1

Let $\alpha$ be an ordinal by Definition 1:

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

$(1)$	$:$		$\alpha$ is a transitive set
$(2)$	$:$		$\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$

where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.

Thus $\alpha$ is a priori transitive.

$\blacksquare$

Proof 2

Let $\alpha$ be an ordinal by Definition 2:

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

$(1)$	$:$	$\alpha$ is a transitive set
$(2)$	$:$	the epsilon relation is connected on $\alpha$:	$\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x $
$(3)$	$:$	$\alpha$ is well-founded.

Thus $\alpha$ is a priori transitive.

$\blacksquare$

Proof 3

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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