Congruence Modulo Zero is Diagonal Relation

Theorem
Congruence modulo zero is the diagonal relation.

That is:
 * $$x \equiv y \left({\bmod\, 0}\right) \iff x = y$$

Proof
Follows directly from the definition of congruence:
 * $$x \equiv y \left({\bmod\, z}\right) \iff x \,\bmod\, z = y \,\bmod\, z$$

When $$z = 0$$ we have by definition:
 * $$x \, \bmod \, z \ \stackrel {\mathbf {def}} {=\!=} \ x$$

And so $$x \,\bmod\, z = y \,\bmod\, z \iff x = z = y$$.

Hence the result.