Let $A_i$ and $B_j$ be sets of the proper cardinality, and we can assume that without loss of generality $A_i\subseteq B_j$ for all $i,j$.
Therefore, $A=\bigcup_{i=1}^m A_i$ is a subset of every $B_j$, and we have that $|A_i|\leq|A|\leq|B_j|$, for every $i,j$.