$X\_i \sim EXP (k\_i)$

I found the pdf of $Y=\min\{X\_1,X\_2\}$:

$$f(y)=(k\_1+k\_2)e^{-y(k\_1+k\_2)}$$

I was told to use conditional probability for finding $P(X\_1 = \min\{X\_1,X\_2\})$

Why can't I just use my pdf, and substitute in $x\_1$ in place of $y$? 

With conditional, would it be:

$P(X\_1<X\_2) = P(X\_1= Y | Y=\min\{X\_1,X\_2\})$ ?