Divisor Relation is Antisymmetric

Theorem
Divides is a antisymmetric relation on $\Z_{>0}$, the set of positive integers.

Corollary
Let $a, b \in \Z$.

If $a \mathop \backslash b$ and $b \mathop \backslash a$ then $a = b$ or $a = -b$.

Proof
We have $\forall a, b \in \Z: a \mathop \backslash b \land b \mathop \backslash a \implies \left\vert{a}\right\vert = \left\vert{b}\right\vert$ which follows from Integer Absolute Value Greater than Divisors:

If we restrict ourselves to the domain of positive integers, we can see:


 * $\forall a, b \in \Z_{>0}: a \mathop \backslash b \land b \mathop \backslash a \implies a = b$

Hence the result.

Proof of Corollary
If $a \mathop \backslash b$ and $b \mathop \backslash a$ then from the main result $\left\vert{a}\right\vert = \left\vert{b}\right\vert$.

The result follows from Every Integer Divides Its Negative and Every Integer Divides Its Absolute Value.