Use the axiom of regularity to demonstrate that if $A$ and $B$ are two sets, then either $A\notin B$ or $B\notin A$ (or both): Proof Assistance
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Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
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Need Hints Prove "$((\neg \alpha \to \alpha) \to \alpha) $" Using Axiom 1,2,3 and MP and deduction theorem
32929
Questions about proving $\lnot \lnot a = a$
62936
Proving equivalencies of two versions of the Axiom of Choice
39708
Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$
46240
Show that if $A$ and $B$ are sets, then $(A\cap B) \cup (A\cap \overline{B})=A$.
41044
Equivalence of $a \rightarrow b$ and $\lnot a \vee b$
48820
Where is axiom of regularity actually used?
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Proving an inclusion related to algebraic sets and interpreting it
62208
If $A$ is a non-empty set, then $A \notin A$
116527
Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
138897
help with some Hilbert style proofs in a propositional logic axiom system.
23764
Proving that $A+A \not\subset 2A $ using a particular definition for sum of sets
144360
For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
32402
$(A \implies B) \implies (\neg B \implies \neg A)$?
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¿Can you help me with axiom of regularity?
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How does the Axiom of Regularity apply to $A=\{1,2,3\}$?