I'll give you two different ways of proving that.
The first makes use of the Distributive Prime Ideal principle:
(DPI) Given a distributive lattice $\mathbf L$ and an ideal $J$ and a filter $G$ of $\mathbf L$ such that $J \cap G = \varnothing$, there exist a prime ideal $I$ and a prime filter $F=L\setminus I$ such that $J\subseteq I$ and $G \subseteq I$.
The (DPI) principle is a weak form of the Axiom of choice (AC).
In fact, it is equivalent to (AC)$_F$, which states that every family of non-empty finite sets has a choice function (see [Davey&Priestley], page 237).
Now, you already have a result stating that a distributive lattice $\mathbf L$ can be embedded in $\wp(X)$, where $X$ is the set of its prime ideals (this is Lemma 10.20 (page 238) in [Davey&Priestley]).
Since $\wp(Y) \cong 2^Y$ for any set $Y$ (just take a subset $A$ to the function $\chi_A:Y\to 2=\{0,1\}$ that makes $\chi_A(y)=1$ iff $y\in A$), the result is proven.
Again, this result is also in [Davey&Priestley], Theorem 10.21.
(Added. Actually we don't need (DPI) in the above proof (indeed, I didn't use it!) because its use must already be incorporated in the proof of the result mentioned by the OP. I misread it as "there is a homomorphism..." when the OP claims "there is a monomorphism..."; (DPI) is useful to prove the homomorphism is one-to-one.)
A different approach would be to show that $\mathbf2$, the two-element lattice is the only (up to isomorphism) distributive lattice which is subdirectory irreducible (and therefore, the variety of distributive lattices is generated by $\mathbf2$) and the fact that lattices are congruence-distributive.
Then, you can apply this result from [Burris&Sankappanavar]:
Corollary 6.10 (Jónsson).
If $\mathcal K$ is a finite set of finite algebras and $V(\mathcal K)$, the variety generated by $\mathcal K$ is congruence-distributive, then the subdirectory irreducible algebras of $V(\mathcal K)$ are in
$$HS(\mathcal K)$$
and
$$V(\mathcal K)=IP_S(HS(\mathcal K)).$$
It follows that $V(\mathbf2)=IP_S(HS(\mathbf2)) \subseteq SP(\mathbf2)$, and so every distributive lattice is a sub-lattice of a power of $\mathbf 2$, i.e, a sub-lattice of $\mathbf2^J$, for some $J$.
(Let me know if you have difficulty with some of the auxiliary results I'm using.)
[Burris&Sankappanavar] S. Burris and H.P. Sankappanavar, A course in Universal Algebra, The Millennium Edition
[Davey&Priestley] B.A. Davey and H.A. Priestley, Introduction to lattices and order, Cambridge University Press, 2nd Edition
Best Answer
I don't know about Bergman, but Burris & Sankappanavar: A Course in Universal Algebra has this to say on page 3: