# Class Difference with Class Difference

## Theorem

Let $A$ and $B$ be classes.

Then:

$A \setminus \paren {A \setminus B} = A \cap B$

## Proof

 $\ds$  $\ds x \in A \setminus \paren {A \setminus B}$ $\ds$ $\leadstoandfrom$ $\ds x \in A \land x \notin \paren {A \setminus B}$ Definition of Class Difference $\ds$ $\leadstoandfrom$ $\ds x \in A \land \lnot \paren {x \in A \land x \notin B}$ Definition of Class Difference $\ds$ $\leadstoandfrom$ $\ds x \in A \land \paren {x \notin A \lor x \in B}$ De Morgan's Laws $\ds$ $\leadstoandfrom$ $\ds \paren {x \in A \land x \notin A} \lor \paren {x \in A \land x \in B}$ Conjunction is Left Distributive over Disjunction $\ds$ $\leadstoandfrom$ $\ds \bot \lor \paren {x \in A \land x \in B}$ Definition of Contradiction $\ds$ $\leadstoandfrom$ $\ds x \in A \land x \in B$ Disjunction with Contradiction $\ds$ $\leadstoandfrom$ $\ds x \in \paren {A \cap B}$ Definition of Class Intersection $\ds$ $\leadsto$ $\ds A \setminus \paren {A \setminus B} = \paren {A \cap B}$ Definition of Class Equality

$\blacksquare$