Intersection of Doubleton

Theorem

Let $x$ and $y$ be sets.

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

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

$\bigcap \set {x, y} = x \cap y$

Proof

\(\ds \)

\(\)

\(\ds z \in \bigcap \set {x, y}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \forall w \in \set {x, y}: z \in w\)

Definition of Intersection of Class

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \forall w: \paren {\paren {w = x \land w = y} \land z \in w}\)

Definition of Doubleton Class

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \paren {z \in x \land z \in y}\)

Equality implies Substitution

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds z \in \paren {x \cap y}\)

Definition of Class Intersection

From Intersection of Non-Empty Class is Set it follows that $x \cap y$ is a set.

$\blacksquare$

Sources

• 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 $5.4. \ \text {(b)}$