A simple proof of the Hilton–Milner theorem

Let [math] be integers and [math] a family of [math] -subsets of [math] . We call [math] intersecting if [math] for all [math] , and we call [math] nontrivial if [math] . Strengthening the famous Erdős–Ko–Rado theorem, Hilton and Milner proved that [math] if [math] is nontrivial and intersecting. We provide a proof by injection of this result.