7. Let the set $S=\{1,2,3, \cdots, 10\}, S$'s subset $A$ satisfies $A \bigcap\{1,2,3\} \neq \varnothing, A \bigcup\{4,5,6\} \neq$ $S$, the number of such subsets $A$ is $\qquad$
$$
A \bigcup\{4,5,6\} \neq S \Leftrightarrow \complement_{S}(A \bigcup\{4,5,6\})=\complement_{S} A \bigcap\{1,2,3,7,8,9,10\} \neq \varnothing,
$$

If $A$ does not contain all elements of the set $\{1,2,3\}$, then the number of subsets $A$ is $\left(2^{3}-2\right) \cdot 2^{7}$; if $A$ contains all elements of the set $\{1,2,3\}$, then the number of subsets $A$ is $2^{3} \cdot\left(2^{4}-1\right)$, so the number of subsets $A$ that satisfy the condition is $\left(2^{3}-2\right) \cdot 2^{7}+2^{3} \cdot\left(2^{4}-1\right)=888$.