How do I say an element is not an element of an element of a set?

Question

So if in the Collatz conjecture $L=\{L_1,\ldots\}$ is the set of all sets of elements of loops and $L_1=\{1,2,4\}$ but we do not know if there exist $L_2,\ldots$ then how do I write that $x$ is not an element of a loop?

Clearly $x\notin L$ is wrong. I guess I need to write $x$ is not in the union of elements of $L$.

Might this be $\cup_{n\in\mathbb{N}}\{L_n\}$ perhaps?

Or $\cup_{L_n\in L}L_n$

Or simply $\cup_L L_n$

Answer 1

So you want to say that $x$ is not in the union of all members of $L$?

$$x~\notin~ \bigcup_{\Lambda\in L}\Lambda$$

This is equivalent to asserting that there is no set in $L$ which contains $x$: $$\forall \Lambda\in L:x \notin \Lambda\\ \neg\exists\Lambda \in L:x \in \Lambda$$