Class Difference of B with Class Difference of A with B

Theorem

Let $A$ and $B$ be classes.

Then:

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

where $S \setminus T$ denotes class difference.

$\ds $

$\ds x \in B \setminus \paren {A \setminus B}$

$\ds $

$\leadstoandfrom$

$\ds x \in B \land x \notin \paren {A \setminus B}$

$\ds $

$\leadstoandfrom$

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

$\ds $

$\leadstoandfrom$

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

$\ds $

$\leadstoandfrom$

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

$\ds $

$\leadstoandfrom$

$\ds x \in B$

$\ds $

$\leadsto$

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

Definition of Class Equality

$\blacksquare$

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