Doubleton of Elements is Subset

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Let $S$ be a set.

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


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


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:

$\set x \cup \set y = \set {x, y}$


$\set {x,y} \subseteq S$


Sufficient Condition

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

From the definition of a subset:

$x \in \set {x,y} \implies x \in S$
$y \in \set {x,y} \implies y \in S$