real analysis – Is a subset contained in a union of its superset?

For context, I’m taking an introductory real analysis course, and our current topic is intervals. One of the questions requires to prove that, for every $x, y$ in some real interval $I$, with $x<y$, the interval $(x,y)$ is also contained in $I$.

I’m taking a shortcut by proving it’s a subset of an open interval $I = (a,b)$, then arguing that $(x,y) subset (I cup G)$, for some arbitrary set $G subset mathbb R$. From there, changing $G$ should span all the cases I’m looking for.

My question: is the provided argument sufficient? Is it a non-trivial fact in set theory? Or am I wrongly using my thesis to prove the hypothesis? Thanks in advance!