10. Given $A_{1}, A_{2}, \cdots, A_{n}$ are $n$ non-empty subsets of the set $A=\{1,2,3, \cdots, 10\}$, if for any $i, j \in$ $\{1,2,3, \cdots, n\}$, we have $A_{i} \cup A_{j} \neq A$, then the maximum value of $n$ is $\qquad$.
Let $m$ be the maximum value of $n$ that satisfies the condition. Since the set $A$ has $2^{10}-1=1023$ non-empty subsets, each subset $A_{i}$ and its complement $\complement_{A} A_{i}$ can be grouped together, resulting in 512 groups, with $A$ being a single group. Clearly, $A$ cannot be taken, so at most 511 non-empty subsets $A_{i}$ can be taken, hence $m \leqslant 511$.
On the other hand, all 511 non-empty subsets of the set $B=\{1,2, \cdots, 9\}$ clearly satisfy the condition, so $m \geqslant 511$. In conclusion, the maximum value of $n$ is 511.