Consider the sets $x_1, x_2,x_3,\ldots$ I was wondering if we could construct a sequence of sets as follows: \begin{equation} x_1=\{x_2\},\;x_2=\{x_3\},\;\ldots \end{equation} and continue to infinity. My initial thoughts were that no, because for any $x\in x_1$, it happens that $x\in x$, given how it's constructed. And this cannot happen because of the separation axiom in ZFC. At least that's my idea.
I just don't know how to put it mathematically, or if the idea is wrong. Could anyone please help?
The Regularity axiom prohibits sequences such as you imagine, more accurately described as running down to infinity. Such a sequence would be an infinite descending $\in$ chain, thus $\{x_n\,|\,n\in\mathbb{N}\}$ would have no $\in$-minimal element.
Notice that if $a = \{b\}$, then $b = \bigcup a$. (The converse does not hold!) Starting with any $x_0$, you can define a sequence of iterated unions:
$$ x_{n+1} = \bigcup x_n. $$ By Regularity, eventually $x_n = \emptyset$, so of course $\emptyset = x_{n+1} \notin x_n = \emptyset$.