How to prove that the counting process of ON periods from a set of $N$ ON-OFF processes is Poissonian?

Question

There are three different and equivalent definitions for the Poisson process (Reference: J. Virtamo slides 38.3143 Queueing Theory / Poisson process):

1. Poisson process is a pure birth process: In an infinitesimal time, interval dt there may occur only one arrival. This happens with the probability $\lambda dt$ independent of arrivals outside the interval.
2. The number of arrivals $N(t)$ in a finite interval of length t obeys the $Poisson(\lambda t)$ distribution, $P\{N(t)=n\}= \frac{(\lambda t)^n}{n!} e^{-\lambda t}$ Moreover, the number of arrivals $N(t_1,t_2)$ and $N(t_3,t_4)$ in non-overlapping intervals $(t_1≤t_2≤t_3≤t_4)$ are independent.
3. The interarrival times are independent and obey the $exp(\lambda)$ distribution: $P\text{{interarrival time$>t$}}=e^{-\lambda t}$

Let us suppose we have $N$ independent ON-OFF processes, i.e., periods of any activity (ON periods) are alternate with inactivity periods (OFF periods). The length of the ON and OFF periods, $T_{ON}$ and $T_{OFF}$, follows any arbitrary distribution and they do not have to be iid. Let us define $X(t)$ as the process that counts the number of ON periods (Let suppose $X(t)$ is incremented at the beginning of the ON period).

When I simulate the $X(t)$ process I always obtain experimentally a “Poisson” process. I mean the interarrival times for the ON periods are very well approximated by an exponential distribution and $X(t)$ with a Poisson distribution. Is there any idea to prove this fact mathematically?

For the moment, I compute the ON periods arrival rate for each individual ON-OFF process as $\lambda_{ON}=\frac{1}{E[T_{ON} ]+E[T_{OFF} ]}$. Then, I consider that the probability that an ON period occurs for a given interval of time $\Delta t$ is $N \cdot \lambda_{ON} \cdot \Delta t$. According to the first definition included for a Poisson process it remains to prove that this probability $\lambda_{ON} \cdot \Delta t$ is independent of arrivals outside the interval. How can I prove it formally? Or it is not necessary to prove anything (is it sufficient to say that the $N$ ON-OFF processes are independent)? I guess that if you consider a single ON-OFF process it is required that $T_{ON}$ and $T_{OFF}$ be distributed exponentially.

Any insight or correction is welcome.