Looking for an algorithm for recognizing finite distributive lattices, I came across Linear Time Recognition Algorithm for Distributive Lattices by Michel Habib and Lhouari Nourine. Just before the beginning of the 2nd chapter, they claim that $$ L\ is\ distributive\Leftrightarrow L\ is\ graded\ and\ |M(L)| = |J(L)| = the\ height\ of\ L$$ where $M(L)$ and $J(L)$ refer to the set of meet-irreducible and the set of join-irreducible elements of $L$, respectively*. They don't, however, provide a proof and I haven't found any mention of a grading function in the referred publication (Dual Symmetry of Projective Sets in a Finite Modular Lattice).
Is this claim about (finite) distributive lattices true, and if so, can you point me toward a proof, or suggest how I would go about proving it?
*Edit: Apparently, their definition of a meet-irreducible (resp. join-irreducible) element is an element of $L$ which is covered by (resp. covers) exactly 1 element. With other definitions it might not work for some lattices, for example the lattice $(\{1,2,3,6,12\}, gcd, lcm)$, with the order of division. In this case, $M(L) = \{2,3,6\}$ and $J(L) = \{2,3,12\}$.