Union of Doubleton
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Theorem
Let $x$ and $y$ be sets.
Let $\set {x, y}$ be a doubleton.
Then $\bigcup \set {x, y}$ is a set such that:
- $\bigcup \set {x, y} = x \cup y$
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
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.7$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Exercise $5.4. \ \text {(a)}$