Rational Number as Power of Number with Itself

Theorem
Every rational number in the interval $\left({\left({\dfrac 1 e}\right)^{\frac 1 e} \,.\,.\, +\infty}\right)$ can be written either as:
 * $a^a$ for some irrational number $a$

or as:
 * $n^n$ for some natural number $n$.