Equivalence of Definitions of Limit Ordinal

Proof
Let $x \in \On$ be an element of the class of all ordinals.

From Categories of Elements under Well-Ordering, $x$ falls into one of the following $3$ categories:


 * $(1): \quad x$ is the smallest element of $\On$ the subset relation


 * $(2): \quad x$ is the immediate successor of another element $y \in \On$ the subset relation


 * $(3): \quad x$ is a limit element of $\On$ under the subset relation.

We have that Zero is Smallest Ordinal.

The result follows.