Limit Ordinals Preserved Under Ordinal Addition

Theorem
Let $x$ and $y$ be ordinals such that $x$ is a limit ordinal.

Then $\left({y + x}\right)$ is a limit ordinal.

That is, letting $K_{II}$ denote the class of all limit ordinals:


 * $\forall x \in K_{II}: \left({y + x}\right) \in K_{II}$

Proof
The result is now obtained by Proof by Contradiction:

Assume that $\left({y + x}\right) = z^+$.

But $w \in x \implies w^+ \in x$ by Successor in Limit Ordinal.

But $z^+ \in z^+$ is clearly a membership loop, and therefore, our assumption must be wrong.

By disjunctive syllogism:
 * $\left({y + x}\right) \in K_{II}$