Limit Ordinals Preserved Under Ordinal Addition

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

Then $y + x$ is a limit ordinal.

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


 * $\forall x \in K_{II}: y + x \in K_{II}$

Proof
The result is now obtained by Proof by Contradiction:

Assume that $y + x = z^+$.

But $w \in x \implies w^+ \in x$ by Successor of Ordinal Smaller than Limit Ordinal is also Smaller.

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

By disjunctive syllogism:
 * $y + x \in K_{II}$