Simple and non-abelian implies perfect

Statement
Suppose $$G$$ is a simple non-abelian group. Then, $$G$$ is a perfect group, i.e., $$G$$ equals its own derived subgroup.

Related facts

 * Characteristically simple and non-abelian implies perfect
 * Simple and non-abelian implies centerless

Facts used

 * 1) uses::Derived subgroup is normal

Proof
Given: A simple non-abelian group $$G$$.

To prove: The derived subgroup $$[G,G]$$ equals $$G$$.

Proof: