Union of Doubleton

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Theorem

Let $x$ and $y$ be sets.

Let $\set {x, y}$ be a doubleton.


Then $\ds \bigcup \set {x, y}$ is a set such that:

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


Proof

\(\ds \) \(\) \(\ds z \in \bigcup \set {x, y}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds \exists w \in \set {x, y}: z \in w\) Definition of Union of Class
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds \exists w: \paren {\paren {w = x \lor w = y} \land z \in w}\) Definition of Doubleton Class
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds \paren {z \in x \lor z \in y}\) Equality implies Substitution
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds z \in \paren {x \cup y}\) Definition of Class Union

Then, from Axiom of Unions, it follows that $x \cup y$ is a set.

$\blacksquare$


Sources