Calculating $P(2\leq X\leq 4)$ for an exponentially random variable

Question

While calculating P(2≤X≤4), for an exponential random distribution, the solution says, $P(2\leq X\leq 4) = F(4)-F(2)$, where F denotes the CDF.

My version is, P(2≤X≤4) = P(22) and P(X≤4), i.e. 1-P(X≤2) and P(X≤4) {1-F(2)} * F(4), presuming they are independent event.

I know it is wrong, but please clarify, where I am commiting mistake in this approach.

Thanks