Working in $\sf ZF - Fnd$, add the following axiom:

**AntiFoundation**: $\forall x: x \neq \emptyset \to \exists! y: y \in y \land y \sim x$

where "$\sim$" stands for *existence of a bijection*.

In this theory we can easily define cardinality as:

$|X|= \kappa \iff \kappa \sim X \land ( \kappa=\emptyset \lor \kappa \in \kappa)$

> The question is if this would imply any kind of Choice? If yes, what would be that form of Choice?

The backround of this question is related to [this][1] question and in particular to [this][2] answer to it, which asserts that some form of Choice would exist even if its for the empty structure (i.e.; Cardinality).


  [1]: https://math.stackexchange.com/q/3774727/489784
  [2]: https://math.stackexchange.com/a/3774771/489784

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