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:


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

Proof
The proof shall proceed by contradiction.

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

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. We are forced to conclude, by disjunctive syllogism, that $( y + x ) \in K_{II}$.