TITLE: How does the axiom of regularity forbid self containing sets?
QUESTION [23 upvotes]: The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem to do nothing about $B=\{B,\emptyset\}$. It is disjoint from $\emptyset$. What is a disproof of the existence of $B$, and how is it related to the axiom of regularity?

REPLY [2 votes]: Here is a formal proof in Fitch: