  Let $G$ be a finite group and let $\pi$ be a set of primes. Write
$\mathrm{Irr}_{\pi'}(G)$ for the set of irreducible characters of degree not
divisible by any prime in $\pi$. We show that if $\pi$ contains at most two
prime numbers and the only element in $\mathrm{Irr}_{\pi'}(G)$ is the principal
character, then $G=1$.
