Lexicographic initial segments preserve ordinary intersection ($|\mathcal{A}|\le \left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$)

For $1\le r\le n/2$, the Erdős-Ko-Rado theorem bounds an intersecting family by the displayed size. The first that many rank-$r$ sets in lexicographic order all contain coordinate one. Therefore the lexicographic initial segment of the same size as any intersecting family is also intersecting. This statement concerns ordinary one-element intersection and does not extend to higher intersection requirements.