Trivial Ordering is Universally Compatible

Theorem
The trivial ordering is an ordering $$\mathcal{R}$$ in a poset $$\left({S; \mathcal{R}}\right)$$ such that:

$$\forall a, b \in S: a \mathcal{R} b \iff a = b$$

That is, there is no ordering defined on any two distinct elements of the set $$S$$.

Proof
To prove that the trivial ordering is in fact an ordering, we need to show that it is:


 * Reflexive, that is, $$\forall a \in S: a \mathcal{R} a$$:

From its definition, we have $$\forall a, b \in S: a = b \Longrightarrow a \mathcal{R} b$$.

Thus, as $$a = a$$, we have $$\forall a \in S: a \mathcal{R} a$$.

So reflexivity is proved.


 * Transitive, that is, $$\forall a, b, c \in S: a \mathcal{R} b \land b \mathcal{R} c \Longrightarrow a \mathcal{R} c$$:

From the definition:


 * $$a \mathcal{R} b \iff a = b$$
 * $$b \mathcal{R} c \iff b = c$$

So as $$a = b \land b = c \Longrightarrow a = c$$ from transitivity of equals, we have that $$a \mathcal{R} c$$ and thus transitivity is proved.


 * Antisymmetric, that is, $$\forall a, b \in S: a \mathcal{R} b \land b \mathcal{R} a \Longrightarrow a = b$$

From the definition:


 * $$a \mathcal{R} b \iff a = b$$.
 * $$b \mathcal{R} a \iff b = a$$.

The result follows from symmetry of equals.