Monotonicity of a convex subdifferential
= Monotonicity of a convex subdifferential
Adding the two <subgradient inequalities> for $p\in\partial f(x)$ and $q\in\partial f(y)$ gives
$$
\langle p-q,x-y\rangle\geq0.
$$
Thus the <subdifferential> is a <monotone operator>. This inequality proves uniqueness of its resolvent: two solutions of $z\in x+\tau\partial f(x)$ must coincide when $\tau>0$.