Ordinal Subset of Ordinal Class

Theorem

Suppose $A$ is an ordinal. Then, $A \subseteq \operatorname{On}$ where $\operatorname{On}$ represents the class of all ordinals.

Proof

By Ordinal Member of Ordinal Class, $A \in \operatorname{On} \lor A = \operatorname{On}$. In either case, $A \subseteq \operatorname{On}$ since $\operatorname{On}$ is transitive.

Source

• Gaisi Takeuti: Introduction to Axiomatic Set Theory (1971):$7.15$