Intersection of two independent sets I am trying to make sure my intuition for the following question from an assignment is correct

Prove or disrove: if $G = (V, E)$ is a graph and $I_1$ and $I_2$ are independent sets in $G$, then $I_1 \cap I_2$ is
an independent set in $G$. 

Answer: This statement is true. Consider arbitrary nodes $x, y \in I_1$ and $u, v \in I _2$. Since $I_1$ and $I_2$ are independent sets in $G$, there is no edge between $\{x, y\}$ and there is no edge between $\{ v, u \}$. Similarly, here is no edge between $\{x, u\}$ and $\{v, y\}$ because otherwise, $I_1$ and $I_2$ would have to have nodes that are adjacent two each other. We see that $I_1 \cap I_2 = \emptyset$ and therefore, the statement that $I_1 \cap I_2$ is
an independent set in $G$ is vacuously true. 

Is this approah necessarily true for any two independent sets in a graph? I would appreciate any feedback
Your proof is unfortunately wrong. Consider what happens when $I_1 = I_2$, for example.

The essential reason behind the statement is the following more general fact:


  Any subset of an independent set is an independent set.


You can try proving this rigorously.