{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nIn many formal set theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], set builder notation is not part of the formal syntax of the theory. Instead, there is a set existence axiom scheme, which states that if $E$ is a set and $\\Phi(x)$ is a formula in the language of set theory, then there is a set $Y$ whose members are exactly the elements of $E$ that satisfy $\\Phi$:\n\n$$(\\forall E)(\\exists Y)(\\forall x)[x \\in Y \\Leftrightarrow x \\in E \\land \\Phi(x)].$$\n\nThe set $Y$ obtained from this axiom is exactly the set described in set builder notation as $\\{ x \\in E \\mid \\Phi(x)\\}$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "inventory": [
      "FB4JJ81R86"
    ],
    "kind": "statement",
    "name": "Set existence axiom",
    "role": "proposition",
    "sources": [
      {
        "locator": "Set-builder notation, section Set existence axiom",
        "mention": "wikipedia://en/220089#IESX",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:220089@r1368739220"
      }
    ],
    "tag": "CP81H8276Z"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/CP81H8276Z/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/CP81H8276Z.md",
  "msc": null,
  "name": "Set existence axiom",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CP8/CP81H8276Z.md",
  "status": null,
  "tag": "CP81H8276Z",
  "uri": "m2://CP81H8276Z",
  "url": "https://math2.tamnd.com/wiki/set-existence-axiom/",
  "writing": {
    "tag": "CP81H8276Z",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}