Equality of Complex Numbers

Theorem
Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers.

Then $z_1 = z_2$ $a_1 = a_2$ and $b_1 = b_2$.

Proof
By definition 2 of a complex number, $z_1$ and $z_2$ can be expressed in the form:
 * $z_1 = \left({a_1, b_1}\right)$
 * $z_1 = \left({a_2, b_2}\right)$

where $\left({a, b}\right)$ denotes an ordered pair.

The result follows from Equality of Ordered Pairs.