# Why don't these assumptions result in further simplification?

I'm trying to simplify an expression, which has a factor of $\lambda(\lambda-1)$, where $\lambda$ may be either $1$, or $0$. So I try:

FullSimplify[λ (λ - 1), λ == 0 || λ == 1]


But the expression doesn't simplify to $0$. I've tried changing the assumptions to λ ∈ Integers && 0 <= λ <= 1, with the same result.

What's going on? How to convince Mathematica to do the simplification?

Simplify[λ (λ - 1), λ (λ - 1) == 0]