Element of a Singleton (set with one element) notation

If you want to be strict with set theoretic context, then $y=\bigcup\{y\}=\bigcup X$. But this might not work very well outside of set theoretic contexts.

In the case that $X$ is a subset of an ordered set, then $y=\min X=\max X$ as well. There's probably no good, and general notation for this. But I honestly don't see why we would need one.

As Asaf said, in contexts (like ZFC) where everything is a set, you can use $\bigcup X$. Unfortnately, I'd expect that only set theorists will recognize what you're doing without further explanation. I've used the notation $\text{TheUnique}(X)$, but that was in a paper close to computer science, where multi-letter symbols like that are fairly common.