 Pair-cover bound for a two-intersecting uniform family

Fix a member of size $r$. Every other member of a two-intersecting family contains a pair from that fixed member. Counting all $r$-sets containing these pairs gives $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{r}{2}\right)\left(\genfrac{}{}{0ex}{}{n-2}{r-2}\right)$ for a uniform set family.