Show that language of distinct binary strings is irregular Let $Σ =\{\textbf{[},\textbf{]},\textbf{,},\textbf{0},\textbf{1}\}$, and let $L⊂Σ^*$ be the language containing list representations of finite sets of binary strings: i.e., every string $x∈L$ is of the form $x= \textbf{[}x_0\textbf{,} x_1\textbf{,} \ldots\textbf{,} x_n\textbf{]}$, where:


for all $0\le i\le n$, $x_i$ is a string in $\{\textbf{0},\textbf{1}\}^*$, and
none of the $x_i$ repeat: if $i$ is not equal to $j$, then $x_i$ is not equal to $x_j$.


Show that $L$ is not regular.

$\
\\$

$\textbf{EDIT: Below is my attempt: }$

Suppose $L$ is regular.

Suppose the pumping length is n.

Consider the string $x$ = $[0^n, 0^{n-1}, ... , 0]$.

By the Pumping Lemma, $x=uvw$, where 


$u$ = $[$
$v$ = $0^n$, $0^{n-1}$, $0$
$w$ = $]$


Then, $|v|&gt;0$ and $|uv| \leq n$

So, $x \in L$ and $|x| \geq n$.

$\
$

Choose $k=2$.

Then, $uv^2w$ = $[0^n, 0^{n-1}, ... , 0  0^n, 0^{n-1}, ... , 0]$

Since each of the strings in x are repeated, this is a contradiction to the Pumping Lemma. Therefore, $L$ is not regular.
Suppose that the pumping length is $n$. Consider the string
$$
w = [0^n,0^{n-1},\ldots,0].
$$
You take it from here.