# Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually:

Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion.

Let me call it Dedekind's lemma. Next to Cayley's theorem:

Every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$.

it is a prominent special case of Yoneda's lemma.

All proofs are constructive, and so is the proof of Erdős' lemma on intersection graphs:

Every graph $G$ is isomorphic to a family of subsets $S_i$ of a set S such that $v_i$ and $v_j$ are joined by an edge iff $S_i \cap S_j \neq \emptyset$