I'd like to write down a proof of the following (simple) fact: $\forall x\left(\bigcup\left\{x\right\}=x\right)$. I'd like to use the rules of inference of natural deduction. One could show that if $y\in\bigcup\left\{x\right\}$ then $y\in x$ 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 $t_1\in\bigcup\left\{t_0\right\}$. Then it follows that $\exists x_2\left(x_2\in\left\{t_0\right\}\land t_1\in x_2\right)$. Now you also suppose that $t_2\in\left\{t_0\right\}\land t_1\in t_2$. So $t_2\in\left\{t_0\right\}$, $t_1\in t_2$, $t_2\in\left\{t_0,t_0\right\}$, $t_2=t_0\lor t_2=t_0$, $t_2=t_0$, $t_1\in t_2\leftrightarrow t_1\in t_0$, $t_1\in t_0$. That's the easy part. The problem now is that if I suppose that $t_1\in t_0$ I can't show that $t_1\in\bigcup\left\{t_0\right\}$.

What can I do? Thanks.