Complex Number equals Conjugate iff Wholly Real

Theorem
Let $z \in \C$ be a complex number.

Let $\overline {z}$ be the complex conjugate of $z$.

Then $z = \overline z$ iff $z$ is wholly real.

Proof
Let $z = x + i y$.


 * Suppose $z = \overline z$.

Then by definition of complex conjugate, $x + i y = x - i y$.

Thus $+y = -y$ and so $y = 0$.

Hence by definition, $z$ is wholly real.


 * Now suppose $z$ is wholly real.

Then $z = x + 0 i = x = x - 0 i = \overline z$.