Every poset is embedded into a meet-semilattice

I "discovered" a few minutes ago that every poset can be embedded into a meet-semilattice.

Let $\mathfrak{A}$ be a poset. Then it is embedded into the meet-semilattice generated by sets $\{ x \in \mathfrak{A} \mid x \le a \}$ where $a$ ranges through $\mathfrak{A}$.

I'm sure I am not the first person who discovered this. Which book could you suggest to read about such things?

This result is mentioned for example as theorem 1.1 in chapter 1 of J.B. Nation's "Revised Notes on Lattice Theory". See also theorem 2.2 in chapter 2. One advantage of Nation's text is that it is freely available.

Simply searching for every poset embedded in Google Books returns some reasonably looking references. (Of course, you can try some other similar search queries.)

For example Theorem 1.11 in the book Steven Roman: Lattices and Ordered Sets uses precisely the embedding you suggested to show that every poset $P$ can be order embedded in a powerset $\mathscr P(P)$.