 Complement splitting for forbidden midpoint intersections

For a prime $p$, take rank-$2p$ subsets of $[4p]$ with no distinct pair meeting in $p$ points. Split by membership of coordinate one. In the containing part, remove that coordinate; in the avoiding part, first take complements and then remove it. Both transformed families have rank $2p-1$ on $4p-1$ points. Their distinct intersections avoid residue $p-1$ modulo $p$. The uniform Frankl-Wilson theorem bounds each by $\left(\genfrac{}{}{0ex}{}{4p-1}{p-1}\right)$, yielding the displayed stronger form of the requested bound.