{
  "aliases": [
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  "body": "\n## Statement\n\nAssume the [[axiom of choice|0HTGB13FE2]].\nLet $(P, \\le)$ be a nonempty [[partially ordered set|011C29Q1S1]] in which each [[chain|0GP9ASVQ1X]] has an [[upper bound|0H5J02P8S9]] in $P$.\nThen $P$ has a [[maximal element|0TZEEFTNCC]].\n\n## Remarks\n\nOver ZF, Zorn's lemma is equivalent to the axiom of choice.\nIf one requires an upper bound also for the empty chain, the hypothesis \"$P$ is nonempty\" is not needed.\nApplications: each vector space has a basis, and each nonzero ring with unity has a maximal ideal.\n\n## History\n\nKazimierz Kuratowski proved the lemma in 1922, and Max Zorn proved it independently in 1935.\nThe [[Hausdorff maximal principle|0KM4KVR9MV]] of 1914 is an earlier statement of the same kind.\n\n## Explanations\n\n### L1 human\n\nZorn's lemma is a tool to show that a maximal object exists.\nYou check that each increasing chain of objects has an object above it.\nThen the lemma gives an object that you cannot enlarge any more.\n",
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