Equivalence of Formulations of Axiom of Choice/Formulation 1 iff Formulation 4

Theorem
The following formulation of the Axiom of Choice:

Formulation 1
implies the following formulation of the Axiom of Choice:

Proof
We note from Set equals Union of Power Set that:
 * $x = \ds \map \bigcup {\powerset x}$

Setting $\powerset A =: s$, we see that from Formulation $1$:


 * $\ds \paren {\O \notin \powerset A \implies \exists \paren {f: \powerset A \to \bigcup \powerset A}: \forall x \in \powerset A: \map f x \in x}$

That is:
 * for every non-empty proper subset of $A$

there exists a mapping $f: \powerset A \to A$ such that:
 * for every non-empty proper subset $x$ of $A$: $\map f x \in x$.

That is Formulation $4$ of the Axiom of Choice