Exponential distribution between events

Question

If I know events happen in intervals of $\Delta T$ seconds between them, where $\Delta T \sim Exp(\frac{1}{\alpha})$, and someone is starting to write down the times in which the events happen at $t=0$, will the first event also be a random variable distributed $Exp(\frac{1}{\alpha})$? It makes sense for it to be the case, but shouldn't the fact that we started writing down the times some time $0<t<\Delta T$ after an unobservable event happened be of any importance?