Ordinal is Transitive/Proof 4
$\alpha$ is an ordinal if and only if:
The proof proceeds by the Principle of Superinduction.
Let $x$ be transitive.
- $x^+$ is transitive.
We have that Class is Transitive iff Union is Subclass.
Hence the result by the Principle of Superinduction.
- 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 $1.7$