Theorem. For a given commutative ring $R$, then $Min(R)$, the set of minimal prime ideals of $R$, is a finite set if and only if no minimal prime ideal of $R$ is contained in the union of the remaining minimal primes.
Sketch of Proof. The implication $\Rightarrow$ of the above nice result is deduced from the prime avoidance lemma. The reverse implication is deduced from the fact that $Min(R)$ is quasi-compact with respect to the flat topology.
Remember that the collection of $V(f)=\{\mathfrak{p}\in Spec(R):f\in\mathfrak{p}\}$ with $f\in R$ forms a sub-base for the opens of the flat topology over $Spec(R)$.