Does the set of all sets containing one set exist? I was reading a pdf on set theory that explains that because of the axiom of regularity, there is no set that contains all sets, $\{x: x =x\}$, because it would have to contain itself.
I was wondering if instead a set containing all sets containing one set, $\{x:\exists y\ x=\{y\}\}$, could exist. How could you prove/disprove it's existence?

No. For any set of sets $\mathcal A, \bigcup \mathcal A = \{x : \exists A \in \mathcal A \text{ with } x \in A\}$ exists (assuming the normal axioms of set theory). But for your supposed set, its union would be the universal set, so your condition does not define a set.
