Union of Doubleton

Theorem

Let $x$ and $y$ be sets.

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

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

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

$\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 $

$\ds \paren {z \in x \lor z \in y}$

$\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.

$\blacksquare$

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 $5.4. \ \text {(a)}$