Let's use the word arrivals for what the OP calls occurrences and let $S_n$ denote the time of the $n$-th arrival. Thus, $S_n$ is a real-valued nonnegative random variable. and it must be that $$S_1 \leq S_2 \leq \cdots \leq S_n \leq S_{n+1} \leq \cdots $$ where the event that any of those $\leq$ signs are actually $=$ signs is an event of probability $0$ as @whuber points out in response to my comment on the main question.
Now, in ordinary probability theory where we begin discussions of probability spaces and what is meant by $(\Omega, \mathcal F, P)$, one of the notions drummed into us is that an event is an element of the $\sigma$-algebra $\mathcal F$ and is thus a specially blessed subset of the sample space $\Omega$ to which the probability measure $P$ assigns a real number called the probability of the event (subject to the axioms etc but no matter). Now, when the experiment is performed, one outcome $\omega \in \Omega$ occurs, but we also say that every event that contains $\omega$ has also occurred -- on any trial of the experiment, a single outcome occurs but multiple events occur. I will likely get down-voted for the following but what the heck: on any trial of the experiment, exactly "half" the events in $\mathcal F$ occur and the other "half" don't in the sense that there is a one-to-one correspondence between events that have occurred and the events that haven't -- one of $A$ and $A^c$ in $\mathcal F$ occurs and the other doesn't, with the reminder to ourselves that $\Omega \in \mathcal F$ always occurs and $\emptyset \in \mathcal F$ never occurs. Note that if event $B$ is a subset (proper subset or not, it doesn't matter) of event $A$ -- that is, $B \subset A$ -- and we are told that $B$ has occurred, then we are can be sure that event $A$ also has occurred: $\omega \in B \implies \omega \in A$.
With this in hand, let us consider the events $C = \{S_n \leq t\}$ and $D = \{S_{n+1} \leq t\}$. If we are told that the event $D$ has occurred, that is, the $(n+1)$-th arrival occurred at or before time $t$, then we know for sure that since $S_n \leq S_{n+1}$, it must be that the $n$-th arrival also occurred at or before the time $t$, that is, the event $C$ also occurred. Event $D = \{S_{n+1} \leq t\}$ is indeed a subset of event $C = \{S_n \leq t\}$.