Simple Events are Mutually Exclusive

Theorem
Let $\EE$ be an experiment.

Let $e_1$ and $e_2$ be distinct simple events in $\EE$.

Then $e_1$ and $e_2$ are mutually exclusive.

Proof
By definition of simple event:

for some elementary events $s_1$ and $s_2$ of $\EE$ such that $s_1 \ne s_2$.

It follows that:

The result follows by definition of mutually exclusive events.