Minimally Inductive Set is Limit Ordinal

Theorem
Let $\omega$ denote the alternative definition of minimal infinite successor set.

Then, $\omega$ is a limit ordinal.

Proof
Let $K_I$ denote the class of all nonlimit ordinals.

Assume $\omega$ is not a limit ordinal. Every element of $\omega$ is a nonlimit ordinal, so if $\omega$ is also a nonlimit ordinal, $\omega + 1 \subseteq K_I$. By the definition of $\omega$, we have $\omega \in \omega$, which is a membership loop and is thus contradictory. Therefore, $\omega$ is a limit ordinal.