자기 자신을 원소로 가지는 집합이 있을까?

Suppose that set \(A\) satisfying \( A \in A\) exists.

Then \( \{A\} \subseteq A\).

Since \(\{A\}\) is set, there is an element \(a \in \{A\}\) such that \( a \cap \{A\}= \emptyset \). (Axiom of Foundation)

\(a \in \{A\}\) implies \(a = A\).

Therefore \( A \cap \{A\} = A = \emptyset\)

But \(A \in A \cap \{A\}\) since \(A \in A\) and \(A \in \{A\}\), deriving contradiction.

To prove a class not exists, we need to approach it by checking axioms. But \(A \in A\) is well-stated, so it's too difficult to show contradiction by handling sets. So I used axiom of foundation to make it element-wise.