If every element of a lattice includes an atom, does it follow that the lattice is atomistic?

Question

Let $L$ be a lattice with no zero element and define '$x$ is an atom' as '$\neg\exists y \, y<x$'. Suppose that $\forall x\exists y (y\text{ is an atom and }y<x)$.

Does it follow that every element in $L$ is a join of atoms?