random variables – Independence of two events, ${S

I was working through J.R. Norris Markov Chains and got stuck at exercise 2.3.1.
The question is the following:

Suppose $S, T$ are independent exponential random variables of parameters $alpha$ and $beta$ respectively. What is the distribution of $min{S,T}$? What is the probability that $S leq T$? Show that the two events ${S < T}$ and ${min{S,T} geq t}$ are independent

I was able to prove:

  1. $min{S,T} sim exp(alpha + beta) $
  2. $mathbb{P}(S leq T) = frac{alpha}{alpha + beta}$

Now for the last problem, I have tried to prove the following ($Z := min{S,T})$
$$
mathbb{P}(Z geq t, S leq T) = mathbb{P}(Z geq t)mathbb{P}(S leq T).
$$

Now the left hand side gives

$$
mathbb{P}(Z geq t, S leq T) = mathbb{P}(min{S,T} geq t mid S leq T)mathbb{P}(S leq T) = mathbb{P}(S geq t)mathbb{P}(S leq T).
$$

But since $mathbb{P}(S geq t) neq mathbb{P}(Z geq t)$, I would conclude that the two events in question are not independent.

My questions are:

  1. What am I doing wrong?
  2. How do you actually solve this question?

Thanks in advance!