Union of Disjoint Singletons is Doubleton/Proof 1
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Theorem
Let $\set a$ and $\set b$ be singletons such that $a \ne b$.
Then:
- $\set a \cup \set b = \set {a, b}$
Proof
Let $x \in \set a \cup \set b$.
Then by the Axiom of Unions:
- $x = a \lor x = b$
It follows from the Axiom of Pairing that:
- $x \in \set {a, b}$
Thus by definition of subset:
- $\set a \cup \set b \subseteq \set {a, b}$
$\Box$
Let $x \in \set {a, b}$.
Then by the Axiom of Pairing:
- $x = a \lor x = b$
So by the Axiom of Unions:
- $x \in \set a \cup \set b$
Thus by definition of subset:
- $\set {a, b} \subseteq \set a \cup \set b$
$\Box$
The result follows by definition of set equality.
$\blacksquare$