Have I properly used $\,\exists !\,$ in this statement?

Question

I want to express the following in logical notation.

For every natural number, there is a unique natural number that succeeds it.

Does the following statement express that proposition?

$\forall n(n\in \mathbb{N} \rightarrow (\exists !m(m\in\mathbb{N} \wedge m=n+1))$

Thank you,

-Hal

Answer 1

Yes, you've done a good job expressing the proposition, using logical notation.

Answer 2

For every natural number, there is a unique natural number that succeeds it.

You could make use of the function notation. In set theoretic notation, we could have $S: \mathbb{N} \to \mathbb{N}$, or (my preference) $\forall x\in \mathbb{N}:S(x)\in \mathbb{N}$.

If $n\in \mathbb{N}$, then $S(n)$ is the unique successor of $n$. The uniqueness of the successor is cleverly built into the notation.