Is this correct way to prove that { $uv$ $\mid$ $u$ and $v$ are strings over $(0,1)$ & $|u| = |v|$ } language is not regular?

Question

I proved this using Pumping Lemma in the following way,

Taking $p$ as pumping length then let us take a string $S$ where, $|u| = |v| = p$

Now according to pumping lemma $S$ can be split into three parts, $$S = xy^{i}z$$

where, $|xy| \le p$ & $|y| \gt 0$.

So now, if $|y| \gt 0$ and taking $i=0$ in $S = xy^{i}z$ will lead to $|u| \lt |v|$ which is not in the language $L$. Thus, this language is not regular.

Is this correct usage of Pumping Lemma?

Edit: This language is regular. See the comments below.