lo.logic - How to demonstrate that the union of the singleton of a set is equal to that set?

I'd like to write down a proof of the following (simple) fact: $forall xleft(bigcupleft{xright}=xright)$. I'd like to use the rules of inference of natural deduction. One could show that if $yinbigcupleft{xright}$ then $yin x$ and vice versa. I managed to show the former implication, but I cannot show the latter.

Let me show you what I accomplished, as an example. You suppose that $t_1inbigcupleft{t_0right}$. Then it follows that $exists x_2left(x_2inleft{t_0right}land t_1in x_2right)$. Now you also suppose that $t_2inleft{t_0right}land t_1in t_2$. So $t_2inleft{t_0right}$, $t_1in t_2$, $t_2inleft{t_0,t_0right}$, $t_2=t_0lor t_2=t_0$, $t_2=t_0$, $t_1in t_2leftrightarrow t_1in t_0$, $t_1in t_0$. That's the easy part. The problem now is that if I suppose that $t_1in t_0$ I can't show that $t_1inbigcupleft{t_0right}$.

What can I do? Thanks.