Symbolizing "There are exactly three..."

Question

Working on the book: P.D. Magnus. "forall x: Calgary. An Introduction to Formal Logic" (p. 233)

24.4 There are exactly...

15. There are exactly three apples.

$A(x): x$ is an apple.

$$ \exists x\exists y\exists z(A(x) \land A(y) \land A(z) \land \lnot(x=y) \land \lnot(x=z) \land \lnot(y=z) \land \forall w(A(w) \to x=w \lor y=w \lor z=w)) $$

Is my symbolisation correct ?

Answer 1

Indeed it is.

"A is satisfied by each from three distinct entities and if any entity satisfies A then it will be one of those three."