Axiom 3.9

Axiom 3.9. (Regularity). 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 \(A\).

Excerpt from wiki: In first order logic, (ignoring urelement), it can be written as \(\forall A (A \neq \emptyset \implies \exists x ( x \in A \land x \cap A = \emptyset))\)

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Exercise 3.2.2

- axiom 3.9: the axiom of regularity

\(\vdash\) { axiom 3.3: \(\exists y ( y \in \{A, B\}) \vdash y = A \lor y = B\) }

\(\vdash\) { axiom 3.9 and modus ponens }

Use the axiom of regularity (and the singleton set axiom) to show that if \(A\) is a set, then \(A \in A\). Furthermore, show that if \(A\) and \(B\) are two sets, then either \(A \notin B\) or \(B \notin A\) (or both).