Doubleton of Elements is Subset
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Theorem
Let $S$ be a set.
Let $\set {x,y}$ be the doubleton of distinct $x$ and $y$.
Then:
- $x, y \in S \iff \set {x,y} \subseteq S$
Proof
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:
- $\set {x,y} \subseteq S$
$\Box$
Sufficient Condition
Let $\set {x,y} \subseteq S$.
From the definition of a subset:
- $x \in \set {x,y} \implies x \in S$
- $y \in \set {x,y} \implies y \in S$
$\blacksquare$