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?