Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. This theorem can be extended as follows: A non-finite distributive lattice that is locally finite and has a $\hat{0}$ is isomorphic to the lattice of finite order ideals of the partial order of the join-irreducible elements of the lattice. In addition, certain ``well ordering'' properties are shown to be equivalent to the premises of the extended theorem.