Using Myhill Nerode to prove that a language is not regular
Let $L$ be your language and let $\equiv$ be the equivalence relation defined by $x \equiv y$ iff $x \in \Sigma^*$ and $y \in \Sigma^*$ have no distinguishing extension (a distinguishing extension is a word $z \in \Sigma^*$ such that exactly one of $xz$ and $yz$ belongs to $L$).
Given $i,j \in \mathbb{N}$ with $i &lt; j$, $0^i$ and $0^j$ cannot belong the same equivalence class since $1^i$ is a distinguishing extension for them: $0^i 1^i \in L$ but $0^j 1^i \not\in L$.
This shows that $L/\mathbin\equiv$ is not finite, and hence $L$ is not regular.