How to prove that minimum of two exponential random variables is another exponential random variable?

Note that you must assume that $X$ and $Y$ are independent, otherwise the result is easily seen to be false.

There is a constant $\lambda$ such that $P(X \geq t)=e^{-\lambda t}$ for every $t>0$.

There is a constant $\mu$ such that $P(Y \geq t)=e^{-\mu t}$ for every $t>0$.

Then for every $t>0$ we have

$$ P(Z \geq t)=P(X\geq t,Y\geq t)=P(X\geq t)P(Y\geq t)=e^{-(\lambda+\mu)t} $$

So $Z$ is an exponential random variable with parameter $\lambda+\mu$.

It might be more intuitive to work with the CDF in this case.

$$F_Z(z) = P(Z < z) = P(\min(X,Y) < z)$$

What is the probability that the minimum of $X$ and $Y$ is below $z$? This will happen if at least one of $X$ and $Y$ is below $z$. The easiest way to deal with probability questions that include the phrase at least one is to find the complementary probability and subtract it from one.

$$P(\text{at least one of $X$ and $Y$ ≤ $z$}) = 1 - P(\text{each of $X$ and $Y$ > z})$$

By independence of $X$ and $Y$ this becomes $1 - P(X > z)P(Y > z)$.

We find $P(X > z) = 1 - F_X(z) = 1 - (1 - e^{-\lambda_X z}) = e^{-\lambda_X z}$ and similarly $P(Y > z) = e^{-\lambda_Y z}$.

Therefore $F_Z(z) = 1 - e^{-\lambda_X z}e^{-\lambda_Y z} = 1 - e^{-(\lambda_X + \lambda_Y) z}$ which is the CDF of an exponential variable with parameter $\lambda_X + \lambda_Y$.

This generalises easily to the case with more than two independent exponential variables.