Doubleton of Elements is Subset

Theorem
Let $S$ be a set.

Let $\set {x,y}$ be the doubleton of distinct $x$ and $y$.

Then:
 * $x, y \in S \iff \set {x,y} \subseteq S$

Necessary Condition
Let $x, y \in S$.

From Singleton of Element is Subset:
 * $\set x \subseteq S$
 * $\set y \subseteq S$

From Union of Subsets is Subset:
 * $\set x \cup \set y \subseteq S$

From Union of Disjoint Singletons is Doubleton:
 * $\set x \cup \set y = \set {x, y}$

Hence:
 * $\set {x,y} \subseteq S$

Sufficient Condition
Let $\set {x,y} \subseteq S$.

From the definition of a subset:
 * $x \in \set {x,y} \implies x \in S$
 * $y \in \set {x,y} \implies y \in S$