9. Let sets $A$ and $B$ satisfy:
$$
A \cup B=\{1,2, \cdots, 10\}, A \cap B=\varnothing \text {. }
$$

If the number of elements in set $A$ is not an element of $A$, and the number of elements in set $B$ is not an element of $B$, then the number of all different sets $A$ that satisfy the conditions is $\qquad$
9. 186 .

Let the number of elements in set $A$ be $k(k=1,2, \cdots, 9)$. Then the number of elements in set $B$ is $10-k$.
According to the problem, $k \notin A, 10-k \notin B, 10-k \in A$. At this point, the number of sets $A$ that meet the requirements is $\mathrm{C}_{10-2}^{k-1}$, where, when $k=5$, it does not satisfy the problem, so $k \neq 5$.
Therefore, the number of sets $A$ is
$$
C_{8}^{0}+C_{8}^{1}+\cdots+C_{8}^{8}-C_{8}^{4}=2^{8}-C_{8}^{4}=186
$$