All explanations and proofs I find about the Pumping Lemma are ambiguous. So if I understand this correctly, if we can find some $p>0$, then for any string $|w| \ge p$, we should be able to split it up into $xyz$ according to some conditions which are satisfied as shown below.
Here's the problem. Say $L$ is the language with equal number of zeros and ones. As far as I know, $L$ is considered to be irregular. But we could easily find $p$ that satisfies the conditions:
$$p=4 $$ $$xyz=0011$$ Let $y$ be the middle part $01$.
$y$ can be pumped.
But what is wrong with this reasoning? It seems to satisfy all the conditions yet still this language is considered irregular.