Logic formula for exactly n unique objects (no more, no less)

Question

I have a question in Logic:

If I am asked to construct a formula, using the '=' predicate, that shows that there are exactly n objects, I need to show that there are no n+1 objects, right?

For example, to show that there are exactly 3 objects, I will show that there are no 4 objects.

Also, I need to show that there exist n objects.

My question is, do I need to show that there is no case where only n-1 objects exist?

For example, to show that there are exactly 3 objects, do I need to show that there is no case in which there are only 2 objects?

If so, is that the correct form to do so? :

---

1) Showing that there is no case of 4 elements:

$ {\lnot}{\exists}x{\exists}y{\exists}z{\exists}w(x{\neq}y{\land}x{\neq}z{\land}x{\neq}w{\land}y{\neq}z{\land}y{\neq}w{\land}z{\neq}w) $

---

2) Showing that there are 3 elements:

$ {\exists}x{\exists}y{\exists}z(x{\neq}y{\land}x{\neq}z{\land}y{\neq}z) $

---

3) Showing that there is no case of only 2 elements:

$ {\lnot}{\exists}x{\exists}y{\lnot}{\exists}z(x{\neq}y{\land}x{\neq}z{\land}y{\neq}z) $

---

And finally, combining the three:

$ (1){\land}(2){\land}(3) $

---

I am really not sure.

Thanks in advance

EDIT:

Actually, assuming that I do need to show that there is no case of less than 3 elements (n elements), I would probably need to show that there is no case where there is just one element as well (from 1 until n-1), am I correct?