Divisor Relation on Positive Integers is Partial Ordering

Theorem
"Divides" is a partial ordering of $$\mathbb{Z}^*_+$$.

Proof
Checking in turn each of the critera for an ordering:

Reflexivity
$$\forall n \in \mathbb{Z}: n \backslash n$$ from Integer Divisor Results‎.

Transitivity
$$\forall x, y, z \in \mathbb{Z}: x \backslash y \land y \backslash z \Longrightarrow x \backslash z$$

This follows because:

Antisymmetry
We have $$\forall a, b \in \mathbb{Z}: a \backslash b \land b \backslash a \Longrightarrow \left|{a}\right| = \left|{b}\right|$$ which follows from Integer Absolute Value Greater than Divisors:

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

$$\forall a, b \in \mathbb{Z}^*_+: a \backslash b \land b \backslash a \Longrightarrow a = b$$

Hence the result.