Class Union Distributes over Class Intersection

Theorem

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

Then:

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

where:

$A \cup B$ denotes class union

$B \cap C$ denotes class intersection.

$\ds $

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

$\ds $

$\leadstoandfrom$

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

Definition of Class Union and Definition of Class Intersection

$\ds $

$\leadstoandfrom$

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

$\ds $

$\leadstoandfrom$

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

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