Equivalence of Formulations of Axiom of Choice

Theorem
The statements of the Axiom of Choice:


 * $(1): \quad$ For every set, we can provide a mechanism for choosing one element of any non-empty subset of the set:


 * $\forall x \in a: \exists P \left({x, y}\right) \implies \exists y: \forall x \in a: P \left({x, y \left({x}\right)}\right)$


 * $(2): \quad$ The Cartesian product of a non-empty family of non-empty sets is non-empty

are equivalent.

Proof
Let $\mathcal C$ be a non-empty set of non-empty sets.

$\mathcal C$ may be converted into an indexed set by using $\mathcal C$ itself as the index and using the identity mapping on $\mathcal C$ to do the indexing.