# A question about axiom of regularity

I am reading Analysis from Terence Tao. The Axiom of regularity is defined as follows:
If A is a non-empty set, then there is at least one element x of A which is either not a set, or is disjoint from A
question: if I have a Set A such as
A:= {A,1}
since 1 is not a set, so axiom of regularity should be hold. But we all know actually it doesn't.

• Consider $A\setminus\{1\}$. This exists if $A$ does, but if it exists it would be a counterexample to regularity. The point is that regularity can constrain us "indirectly" - $X$ might be "bad" because it lets us construct $Y$ which violates regularity. Jul 9, 2022 at 14:54
• This question should contain more context. For example, In some contexts, a set can contain itself, whereas in other’s (ZFC for example) it cannot. In other words, whether or not your definition of $A$ is acceptable depends on the context. Jul 9, 2022 at 15:03
• @NoahSchweber Why is a counterexample of regularity? {{A}} and {A} are still disjoint right? sorry i might be dumb. Jul 9, 2022 at 17:16
• Let $B = A \setminus \{1\} = \{A\}$. Then $B$ contains at least one element, contains no elements that are not a set, and all its elements intersect it, as $A \cap B = \{A, 1\} \cap \{A\} = \{A\}$. Jul 9, 2022 at 18:58
• @mihaild thanks! now i'm clear. Jul 9, 2022 at 19:44