Poisson Process as Exponential Interarrivals

Question

I was trying to understand the derivation of the poisson process as a counting process with exponential interarrival times, and came upon this source. They've defined $S_n$ as the time of the $n$th arrival and $T_n$ as the $n$th interarrival time. While most things make sense to me, I'm very confused as to how they went from $P[S_{n+1} > t \cap (\bar{S_n} > t)]$ to $P[S_{n+1} > t] - P[S_n > t]$, and would appreciate some clarification. Thank you!

Answer 1

$\{S_n > t\} \subseteq \{S_{n+1} > t\}$ because if the $n$th arrival is after time $t$, the $(n+1)$st arrival is also after time $t$.

Thus, $\{S_{n+1} > t\} \cap (\text{not } \{S_n > t\})$ are the outcomes in $\{S_{n+1} > t\}$ that are not in $\{S_n > t\}$; this difference has probability $P(S_{n+1} > t) - P(S_n > t)$.