Can union notation be simplified

Question

Just for context consider something like a set $\lbrace X_i \rbrace_{i \in I}$ where $I$ is also a set. I must say I'm more familiar with the sequencial notation $(X_i)_{i \in I}$ but I guess this makes it less of a set-related question. Should we need to refer to the set whose elements are the elements of the elements of the set $\lbrace X_i \rbrace_{i \in I}$ we would use the notation $\cup_{i\in I} X_i$. However, if the set $\lbrace X_i \rbrace_{i \in I}$ was given a name, like $S$, then we would write $\cup_{s\in S} s$. But then why not write $\cup_S$ ? It seems totally unambiguous to me. I'm inclined to think that no extra information is given by that $s$ in the indices since it is $s$ itself that plays the role of the elements to be united, and not a function of $s$. Is this notation ever used ?