Ordinal is Transitive/Proof 4

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Every ordinal is a transitive set.


Let $\alpha$ be an ordinal by Definition 4.

$\alpha$ is an ordinal if and only if:

$\alpha$ is an element of every superinductive class.

The proof proceeds by the Principle of Superinduction.

From Empty Class is Transitive we start with the fact that $0$ is transitive.


Let $x$ be transitive.

From Successor Set of Transitive Set is Transitive:

$x^+$ is transitive.


We have that Class is Transitive iff Union is Subclass.

Hence the union of a chain of transitive sets is transitive.


Hence the result by the Principle of Superinduction.