Constructing a sequence up to infinity
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 $\emptyset = x_{n+1} \notin x_n = \emptyset$.