{
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  "body": "\n## Statement\n\nAssume the [[axiom of choice|0HTGB13FE2]].\nLet $A$ and $B$ be [[sets|0SJ3RX9TXA]].\nThen there is an [[injective function|0PVG1CRAJN]] from $A$ to $B$ or an injective function from $B$ to $A$.\nSo $|A| \\le |B|$ or $|B| \\le |A|$.\n\n## Remarks\n\nWith the [[Schröder–Bernstein theorem|0A9V6RKQG6]], it follows that the [[order of cardinals|01GTP8M5RG]] is a total order.\nThe statement is equivalent to the axiom of choice over ZF.\nOne proof uses [[Zorn's lemma|04RXTQA5ZA]] on the set of injective partial functions from $A$ to $B$.\n\n## Explanations\n\n### L1 human\n\nAny two sets can be compared in size: one of them fits into the other.\nWithout the axiom of choice, two sets can exist such that neither fits into the other.\n",
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