Sets That Avoid Their Own Size

Nonempty sets $A$ and $B$ satisfy $A \cup B = \{1, 2, 3, \dots, 10\}$ and $A \cap B = \varnothing$. The number of elements of $A$ is not an element of $A$, and the number of elements of $B$ is not an element of $B$. Find the number of possible sets $A$.