Additive and Multiplicative Inverses Are Unique

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $r$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring.

¶ For every $x\in r$ there is exactly one $w\in r$ with $x+w=0$.

¶ If $r$, with $+$, $\cdot$, $0$ and $1$, is a field, then for every $x\in r$ with $x\neq0$ there is exactly one $y\in r$ with $x\cdot y=1$.