Suppose first that $\overline{z} = \overline{(x, y)} = (0, 0)$. Since $(0, 0)$ has its second component zero, then results

(i)	R2950: Complex conjugation operation is an involution
(ii)	R2413: Complex conjugate of real number

imply \begin{equation} z = \overline{\overline{z}} = \overline{(0, 0)} = (0, 0) \end{equation} Conversely, assume that $z = (0, 0)$. Then result R2413: Complex conjugate of real number immediately implies $\overline{z} = \overline{(0, 0)} = (0, 0)$. $\square$