Class Intersection Distributes over Class Union

Theorem

Let $A$, $B$ and $C$ be classes.

Then:

$A \cap \paren {B \cup C} = \paren {A \cap B} \cup \paren {A \cap C}$

where:

$A \cap B$ denotes class intersection

$B \cup C$ denotes class union.

$\ds $

$\ds x \in A \cap \paren {B \cup C}$

$\ds $

$\leadstoandfrom$

$\ds x \in A \land \paren {x \in B \lor x \in C}$

Definition of Class Union and Definition of Class Intersection

$\ds $

$\leadstoandfrom$

$\ds \paren {x \in A \land x \in B} \lor \paren {x \in A \land x \in C}$

$\ds $

$\leadstoandfrom$

$\ds x \in \paren {A \cap B} \cup \paren {A \cap C}$

Definition of Class Union and Definition of Class Intersection

$\blacksquare$

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 $5.6. \ \text {(a)}$