Example 9 For a finite set $A$, function $f: N \rightarrow A$ has only the following property: if $i, j \in N, |H| i-j |$ is a prime number, then $f(i) \neq f(j)$. How many elements does set $A$ have at least?
Solution: Let $|A|$ denote the number of elements in the finite set $A$, and estimate the lower bound of $|A|$.

Since the absolute value of the difference between any two numbers among $1, 3, 6, 8$ is a prime number, by the problem's condition: $f(1)$, $f(3)$, $f(6)$, $f(8)$ are four distinct elements in $A$. Therefore, $|A| \geqslant 4$.

Construct the set $A=\{0,1,2,3\}$, then the function $f: N \rightarrow A$ has the following correspondence:
. If $x \in N, x=4 k+r$, then $f(x)=r$. That is, $k \in N \bigcup\{0\}, r=0,1,2,3$.
Next, we prove that $f$ satisfies the conditions of the problem:
For any $x, y \in N$, if $|x-y|$ is a prime number, then $f(x) \neq f(y)$. Otherwise, if $f(x)=f(y)$, then
$x \equiv y(\bmod 4)$.

Thus, $4 \mid |x-y|$, which contradicts the fact that $|x-y|$ is a prime number.
From the above, we conclude that $|A|_{\text {min }}=4$.
The proof of the inequality involves various techniques, such as constructing counterexamples, constructing sequences, constructing propositions, etc. Please list them.