It is usually said that the Russell set (the set $R$ of all sets that are not members of themselves) is non-well-founded, but I honestly do not understand why.
If an object is non-well-founded, it should somehow violate the Axiom of Regularity below (which is sometimes called Foundation Axiom, such as in https://plato.stanford.edu/entries/nonwellfounded-set-theory/#2.2):
Axiom of Regularity: For every non-empty set $x$, there is some $y \in x$ such that $y\cap x=\varnothing$.
But I cannot see how exactly the Russell set violates the axiom. Actually, it seems to respect it: since the empty set does not contain itself, it is a member of the Russell set $R$, and so there is a $y \in R$ such that $y\cap R=\varnothing$, namely, the empty set $\varnothing$. So what do people mean when they say that the Russell set is non-well-founded? Can someone guide me through the strategy to prove this?
Observation: I've read the discussion in Non WellFounded Set theories and Russell's Paradox and it was very helpful, but I still don't understand this particular point. Noah, in his answer, said this: "Let $𝑋$ be the set of all well-founded sets. Note that by regularity this is equal to the set of all sets not containing themselves." This must mean that the axiom of regularity is equivalent to the statement "a set cannot be a member of itself," but I don't understand why. I'm sure I'm missing something here, but I don't know what.
Thanks a lot!