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  "body": "\n## Statement\n\nAssume the [[axiom of choice|0HTGB13FE2]].\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nThen each [[chain|0GP9ASVQ1X]] of $P$ is a subset of a maximal chain.\nA maximal chain is a chain that is not a proper subset of another chain of $P$.\n\n## Remarks\n\nOver ZF, the principle is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].\nWith the empty chain, it says that each partially ordered set has a maximal chain.\nFelix Hausdorff proved it in 1914.\n\n## Explanations\n\n### L1 human\n\nStart with any chain in the order.\nYou can extend it until no element can be added without breaking the chain property.\n",
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