B-Algebra Identity: xy = 0 iff x = y

Theorem
Let $\struct {X, \circ}$ be a $B$-algebra.

Then:


 * $\forall x, y \in X: x \circ y = 0 \iff x = y$

Sufficient Condition
Suppose that $x = y$.

Then by Axiom $(\text A 1)$ for $B$-algebras:


 * $x \circ y = x \circ x = 0$

Necessary Condition
Let $x, y \in X$ such that $x \circ y = 0$.

Then:

Hence the result.