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This is a follow up on this earlier question of mine.

We have the following statements:

(HSO) For every infinite set $X$ there exists an injection $f: X \times X \hookrightarrow X$

(HSU) For every infinite set $X$ there exists an injection $f: [X]^2 \hookrightarrow X$

One can prove (HSO) $\rightarrow $ (HSU) as follows:

Assume (HSO). Since $h: X \to X \times X$, $x \mapsto (x,x)$ is an injection by (HSO) and Cantor-Schröder-Bernstein, $X \approx X \times X$ for every infinite set $X$. By a result of Tarski, the latter implies the Axiom of Choice. Once we have AC we can prove HSU as follows: let $f: X \times X \to X$ be an injection. For each unordered pair $\{x,y\}$ choose $g(\{x,y\})$ from the set $\{f((x,y)), f((y,x))\}$.

The following is an exercise I have been unable to solve:

Exercise 40: Find a simpler proof of the implication (HSO) $\rightarrow$ (HSU) in ZF.

Can someone help me solve this exercise? Thanks for your help.

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Thanks for the upvote that reminded me of this question! –  Matt N. Jun 6 '13 at 5:20
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