Minimal normal implies pi-group or pi'-group in pi-separable group

Statement

Suppose $\pi$ is a set of primes and $G$ is a finite group that is $\pi$-separable (see Pi-separable group (?)). Suppose $H$ is a Minimal normal subgroup (?) of $G$. Then, $H$ is either a $\pi$-group (i.e., all prime factors of the order of $H$ are in $\pi$) or a $\pi '$-group (i.e., none of the prime factors of the order of $H$ are in $\pi$).

Related facts

• Minimal normal implies central in nilpotent
• Minimal normal implies elementary abelian in finite solvable