The goal is to prove that the axiom of choice is implied by the statement "for every infinite set ${\displaystyle A:}$  ${\displaystyle |A|=|A\times A|}$ ". It is known that the well-ordering theorem is equivalent to the axiom of choice; thus it is enough to show that the statement implies that for every set ${\displaystyle B}$  there exists a well-order.

Since the collection of all ordinals such that there exists a surjective function from ${\displaystyle B}$  to the ordinal is a set, there exists an infinite ordinal, ${\displaystyle \beta ,}$  such that there is no surjective function from ${\displaystyle B}$  to ${\displaystyle \beta .}$  We assume without loss of generality that the sets ${\displaystyle B}$  and ${\displaystyle \beta }$  are disjoint. By the initial assumption, ${\displaystyle |B\cup \beta |=|(B\cup \beta )\times (B\cup \beta )|,}$  thus there exists a bijection ${\displaystyle f:B\cup \beta \to (B\cup \beta )\times (B\cup \beta ).}$ 

For every ${\displaystyle x\in B,}$  it is impossible that ${\displaystyle \beta \times \{x\}\subseteq f[B],}$  because otherwise we could define a surjective function from ${\displaystyle B}$  to ${\displaystyle \beta .}$  Therefore, there exists at least one ordinal ${\displaystyle \gamma \in \beta }$  such that ${\displaystyle f(\gamma )\in \beta \times \{x\},}$  so the set ${\displaystyle S_{x}=\{\gamma :f(\gamma )\in \beta \times \{x\}\}}$  is not empty.

We can define a new function: ${\displaystyle g(x)=\min S_{x}.}$  This function is well defined since ${\displaystyle S_{x}}$  is a non-empty set of ordinals, and so has a minimum. For every ${\displaystyle x,y\in B,x\neq y}$  the sets ${\displaystyle S_{x}}$  and ${\displaystyle S_{y}}$  are disjoint. Therefore, we can define a well order on ${\displaystyle B,}$  for every ${\displaystyle x,y\in B}$  we define ${\displaystyle x\leq y\iff g(x)\leq g(y),}$  since the image of ${\displaystyle g,}$  that is, ${\displaystyle g[B]}$  is a set of ordinals and therefore well ordered.

• Rubin, Herman; Rubin, Jean E. (1985), Equivalents of the Axiom of Choice II, North Holland/Elsevier, ISBN 0-444-87708-8
• Mycielski, Jan (2006), "A system of axioms of set theory for the rationalists" (PDF), Notices of the American Mathematical Society, 53 (2): 209
• Tarski, A. (1924), "Sur quelques theorems qui equivalent a l'axiome du choix", Fundamenta Mathematicae, 5: 147–154, doi:10.4064/fm-5-1-147-154