Detail about an answer of Any open subset of $\mathbb{R}$ is a countable union of disjoint open intervals

I saw an answer to the question that any open subset of Ris a countable union of disjoint open intervals.The link of the beautiful answer is this.I understand the statement in the answer but it remains one question: how to prove $I_{q}$ and $I_{p}$ are disjoint?The answerer says in comment that if $x \in I_{p} \cap I_{q}$, then $I_{p} \cup I_{q} \subseteq I_{p},I_{q}$ according to the definition of $I_{p}$, hence $I_{p} \cap I_{q} \ne \emptyset$ implies $I_{p} = I_{q}$. I have no idea why is it like that. Any help will be greatly appreciated.