Characters of ${\pi }^{\mathrm{\prime }}$-degree

Let $G$ be a finite group and let $\pi$ be a set of primes. Write ${\mathrm{I}\mathrm{r}\mathrm{r}}_{{\pi }^{\mathrm{\prime }}}(G)$ 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 ${\mathrm{I}\mathrm{r}\mathrm{r}}_{{\pi }^{\mathrm{\prime }}}(G)$ is the principal character, then $G=1$.