First of all, of course integers, have multiplicative inverses and of course that statement is obviously false as $a = \frac 12, \frac 13, \frac 14,$ etc. bear out.
What you mean to say is $1$ and $-1$ are the only two integers whose multiplicative inverses are also integers.
I.e. If $a, \frac 1a \in \mathbb Z$ then $a = 1$ or $a = -1$.
That is not an axiom.
If $a \in \mathbb Z$ then you have five possibilities. $a > 1$ or $a = 1$ or $a = 0$ or $a = -1$ or $a < 1$.
Case 1: $a > 1$. Then if $\frac 1a < 0$ we would have $a*\frac 1a < a*0$ and $1 < 0$ which is a contradiction. We can't have $\frac 1a = 0$ because $0$ has no multiplicative inverses so $\frac 1a > 0$. If $\frac 1a \ge 1$ then $a*\frac 1a \ge a*1 = a$ or $1 \ge a$ which is a contradiction so $0< \frac 1a < 1$ and not an integer.
Case 2: $a = 1$. then $\frac 1a = 1$.
Case 3: $a = 0$ then $\frac 1a$ is undefined.
Case 4: $a = -1$ then $\frac 1a = -1$.
Case 5: $a < -1$. that is so similar to case 1: I'll leave it to you.