Show that K is a subfield of F if and only if $\displaystyle 0\neq a\in K$ implies that $\displaystyle a^{-1}\in K$

So, we know that all fields have inverses. So, if $\displaystyle 0\neq a$ then we can say $\displaystyle a=n\in K$. Then we have $\displaystyle n+(-n)=0$ and $\displaystyle nn^{-1}=1=n^{-1}n$ because $\displaystyle a\neq 0$. Now this doesn't completely show $\displaystyle \rightarrow$ part of the iff does it? If we know that the element has an inverse, can we say it satisfies all of the other axioms of a field?