A generalization of the Arad–Ward theorem on Hall subgroups

For a set of primes π, denote by $E_{\pi }$ the class of finite groups containing a Hall π-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 π is a set of primes, l is an integer such that $2⩽l<\left|\pi \right|$ and G is a finite group that contains a Hall ρ-subgroup for every subset ρ of π of size l, then G contains a solvable Hall π-subgroup.