Minimal spectrum of a commutative ring

Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I remember that this is true but can't seem to prove it myself or find the proof anywhere.

I would be especially happy if there is some proof that does not use the (easy) fact that this space is Hausdorff. This is because I am trying to prove that the primitive spectrum of a certain noncommutative ring, which I know is not Hausdorff, is totally disconnected. Hopefully the proof works in my situation when phrased in the appropriate way.

Let $\text{MinSpec}(A) \subset \text{Spec}(A)$ be the subset of minimal primes. I claim that $\text{MinSpec}(A) \cap W$ is open and closed whenever $W$ is a quasi-compact open of $\text{Spec}(A)$. This follows immediately from Lemma Tag 00EV (which has a purely algebraic proof). Hence $\text{MinSpec}(A)$ has a base for its topology consisting of closed and open subsets. Thus it is totally disconnected.