 Proof: Like completeness, this proof is complex, and we rest content with a sketch. The downward Löwenheim-Skolem theorem invokes the axiom of choice, and indeed, is equivalent to the axiom of choice (see the entry on  the axiom of choice).  So let \(C\) be a choice function on the powerset of \(d\), so that for each non-empty subset \(e\subseteq d, C(e)\) is a member of \(e\). We stipulate that if \(e\) is the empty set, then \(C(e)\) is \(C(d)\).