{
  "aliases": [
    "cantor-bernstein-theorem",
    "cantor-schroeder-bernstein-theorem"
  ],
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  "body": "\n## Statement\n\nLet $A$ and $B$ be [[sets|0SJ3RX9TXA]].\nIf there are [[injective functions|0PVG1CRAJN]] $f \\colon A \\to B$ and $g \\colon B \\to A$, then there is a [[bijective function|0SNQEF7Z53]] $h \\colon A \\to B$.\n\n## Remarks\n\nIn terms of cardinals: $|A| \\le |B|$ and $|B| \\le |A|$ imply $|A| = |B|$. So the [[order of cardinals|01GTP8M5RG]] is antisymmetric.\nThe proof does not use the axiom of choice.\nExample: $[0, 1]$ and $[0, 1)$ are equinumerous, because each one injects into the other.\n\n## History\n\nCantor stated the theorem in 1887 without a proof.\nDedekind proved it in 1887, but he did not publish the proof.\nBernstein and Schröder published proofs in 1898, and the proof of Schröder had a flaw.\n\n## Explanations\n\n### L1 human\n\nIf $A$ fits into $B$ and $B$ fits into $A$, then $A$ and $B$ have the same size.\nFor finite sets this is obvious, but for infinite sets it needs a clever construction of $h$.\n",
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    "aliases": [
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  "msc": [
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  "name": "Schröder–Bernstein theorem",
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