Union of Doubleton

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Then:

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


Proof

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

$\blacksquare$


Sources