Ring Zero is Unique/Proof 3

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Then the ring zero of $R$ is unique.

Proof
Suppose $0$ and $0\,'$ are both ring zeroes of $\left({R, +, \circ}\right)$.

Then by Ring Product with Zero:
 * $0\,' \circ 0 = 0$ by dint of $0$ being a ring zero
 * $0\,' \circ 0 = 0\,'$ by dint of $0\,'$ being a ring zero.

So $0 = 0\,' \circ 0 = 0\,'$.

So $0 = 0\,'$ and there is only one ring zero of $\left({R, +, \circ}\right)$ after all.