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  "body": "\n## Proof\n\nWe apply Zorn's lemma to the partial injections from $A$ to $B$.\n\n1. Let $A$ and $B$ be sets. A partial injection is an [[injective function|0PVG1CRAJN]] $p \\colon D \\to B$ with $D \\subseteq A$. We view $p$ as its set of pairs $\\{(a, p(a)) : a \\in D\\}$.\n2. Let $Q$ be the set of partial injections, ordered by inclusion $\\subseteq$. This is a [[partial order|011C29Q1S1]]. The empty function lies in $Q$, so $Q$ is nonempty.\n3. Let $K$ be a [[chain|0GP9ASVQ1X]] of $Q$. If $K$ is empty, the empty function is an [[upper bound|0H5J02P8S9]] of $K$.\n4. Let $K$ be nonempty, and let $q = \\bigcup_{p \\in K} p$ be the [[union|05GK32H72H]] of these sets of pairs.\n5. Let $(a, b), (a', b') \\in q$, with $(a, b) \\in p$ and $(a', b') \\in p'$ for $p, p' \\in K$. Since $K$ is a chain, both pairs lie in one member $r \\in K$, the larger of $p$ and $p'$.\n6. If $a = a'$, then $b = b'$, since $r$ is a [[function|0XHDTE5YMZ]]. If $b = b'$, then $a = a'$, since $r$ is injective.\n7. By step 6, $q$ is an injective function from a subset of $A$ to $B$. So $q \\in Q$, and $q$ is an upper bound of $K$.\n8. By [[Zorn's lemma|04RXTQA5ZA]], $Q$ has a [[maximal element|0TZEEFTNCC]] $m \\colon D \\to B$. Let $E = m(D)$ be its image.\n9. Suppose that $D \\neq A$ and $E \\neq B$. Pick $a \\in A \\setminus D$ and $b \\in B \\setminus E$.\n10. Then $m \\cup \\{(a, b)\\}$ is a function on $D \\cup \\{a\\}$, and it is injective, because $b \\notin E$. It lies in $Q$ and strictly contains $m$, contradicting maximality.\n11. So $D = A$ or $E = B$. If $D = A$, then $m \\colon A \\to B$ is injective.\n12. If $E = B$, then $m \\colon D \\to B$ is a bijection. Its inverse $m^{-1} \\colon B \\to D \\subseteq A$ is an injective function from $B$ to $A$.\n13. So there is an injection from $A$ to $B$ or from $B$ to $A$. By the [[order of cardinals|01GTP8M5RG]], $|A| \\le |B|$ or $|B| \\le |A|$.\n\n## Idea\n\nBuild an injection from $A$ to $B$ as far as possible. Zorn's lemma gives one that cannot be extended. It cannot be extended only if it has used up all of $A$ or all of $B$, and each case gives an injection in one direction.\n",
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