Congruence Modulo Zero is Diagonal Relation

Theorem

Congruence modulo zero is the diagonal relation.

That is:

$x \equiv y \pmod 0 \iff x = y$

Proof

Follows directly from the definition of congruence:

$x \equiv y \pmod z \iff x \bmod z = y \bmod z$

When $z = 0$ we have by definition:

$x \bmod 0 := x$

And so $x \bmod 0 = y \bmod 0 \iff x = y$.

Hence the result.

$\blacksquare$