Ordinal is Transitive/Proof 4

Theorem

Every ordinal is a transitive set.

Proof

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.

$\Box$

Let $x$ be transitive.

From Successor Set of Transitive Set is Transitive:

$x^+$ is transitive.

$\Box$

We have that Class is Transitive iff Union is Subclass.

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

$\Box$

Hence the result by the Principle of Superinduction.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.7$