Example 1 (1998 National High School League Question) If the non-empty set $A=\{x \mid 2 a+1 \leqslant x \leqslant 3 a-5\}, B=\{x \mid 3 \leqslant x \leqslant 22\}$, then the set of all $a$ that makes $A \subseteq(A \cap B)$ true is ( ).
A. $\{a \mid 1 \leqslant a \leqslant 9\}$
B. $\{a \mid 6 \leqslant a \leqslant 9\}$
C. $\{a \mid a \leqslant 9\}$
D. $\varnothing$
Solution: Choose B. Reason: According to the definition of intersection, $(A \cap B) \subseteq A$, and it is also given that $A \subseteq (A \cap B)$, thus we can conclude $A = A \cap B$, which means $A \subseteq B$.
As shown in Figure 5-1, for $A \subseteq B$ to hold, the following conditions must be satisfied:
\[
\left\{\begin{array}{l}
2a + 1 \geqslant 3, \\
3a - 5 \leqslant 22, \\
3a - 5 \geqslant 2a - 1.
\end{array}\right.
\]

Solving these, we get $6 \leqslant a \leqslant 9$.