Example 1 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) $\varnothing$
(1998, National High School Mathematics Competition)
Solution: According to
$$
A \subseteq(A \cap B)
$$

we know $A \subseteq B$,
as shown in Figure 1.
Thus,
$$
\left\{\begin{array}{l}
2 a+1 \geqslant 3, \\
3 a-5 \leqslant 22, \\
3 a-5 \geqslant 2 a+1
\end{array}\right.
$$
$\Rightarrow 6 \leqslant a \leqslant 9$, i.e., $a \in\{a \mid 6 \leqslant a \leqslant 9\}$.
Therefore, the answer is (B).
Note: Using a Venn diagram or a number line can intuitively represent the relationship between sets, making the problem clearer and more understandable.