= Solution
Fix $A\in\mathcal F$ and consider the traces
$$
\mathcal T=\{A\cap B:B\in\mathcal F\setminus\{A\}\}\subseteq\mathcal P(A).
$$
For distinct $B,C\in\mathcal F\setminus\{A\}$, the hypothesis applied in both orders says
$$
A\cap B\not\subseteq C,
\qquad
A\cap C\not\subseteq B.
$$
Equivalently, neither trace contains the other. Thus the traces are distinct and form an <antichain> in the Boolean lattice on the $2r$ points of $A$. By <Sperner theorem>,
$$
|\mathcal F|-1=|\mathcal T|\leq\binom{2r}{r},
$$
which is the required bound.
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