Simple Graph with Finite Vertex Set is Finite

Theorem
Let $G$ be a simple graph.

Suppose that the vertex set of $G$ is finite.

Then $G$ is a finite graph.

That is to say, its edge set is also finite.

Proof
Since $G$ is simple, it can have at most one edge between each two vertices.

Therefore, the mapping:


 * $v: E \left({G}\right) \to \mathcal P \left({V \left({G}\right)}\right), v \left({e}\right) = \left\{{a_e, b_e}\right\}$

which assigns to each edge its endvertices $a_e, b_e$, is an injection.

From Power Set of Finite Set is Finite and Domain of Injection Not Larger than Codomain, it follows that $E \left({G}\right)$ is finite.

Hence $G$ is finite.