A generalization of the Arad--Ward theorem on Hall subgroups

For a set of primes $\pi$, denote by $E_{\pi }$ the class of finite groups containing a Hall $\pi$-subgroup. We establish that $E_{{\pi }_1}\cap E_{{\pi }_2}$ is contained in $E_{{\pi }_1\cap {\pi }_2}$. As a corollary, we prove that if $\pi$ is a set of primes, $l$ is an integer such that $2⩽l<\mathrm{\mid }\pi \mathrm{\mid }$ and $G$ is a finite group that contains a Hall $\rho$-subgroup for every subset $\rho$ of $\pi$ of size $l$, then $G$ contains a solvable Hall $\pi$-subgroup.