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  "body": "\n## Definition\n\nWork in first-order logic with the relation symbol $\\in$.\nThe *axiom of regularity*, or *axiom of foundation*, is the axiom\n$$\\forall x\\, \\bigl( x \\ne \\emptyset \\to \\exists y\\, ( y \\in x \\wedge y \\cap x = \\emptyset ) \\bigr).$$\nIn words: every nonempty [[set|0SJ3RX9TXA]] $x$ has an [[element|0EZ758XE73]] $y$ such that no element of $y$ is an element of $x$.\n\n## Remarks\n\nExample: let $x = \\set{\\emptyset, \\set{\\emptyset}}$.\nThe element $y = \\emptyset$ satisfies $y \\cap x = \\emptyset$, because the [[empty set|0ECVK59Y03]] has no elements.\nNon-example: a set $a$ with $a = \\set{a}$ would violate the axiom.\nThe only element of $x = \\set{a}$ is $a$, and $a \\cap x = x \\cap x = x \\ne \\emptyset$.\nSo in [[ZF|0HE5WDQ9FJ]], no set is an element of itself.\nThe axiom says that $\\in$ is a [[well-founded relation|0230ECX1Q5]] on every set.\nOver the other axioms of ZF, it is equivalent to the statement that every set lies in some level $V_\\alpha$ of the cumulative hierarchy.\nWith the axiom of dependent choice, it is equivalent to the statement that there is no sequence $(x_n)$ with $x_{n+1} \\in x_n$ for all $n$.\nIn Mathlib, the statement for ZFC sets is `ZFSet.regularity`.\n",
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