I believe I've encountered the statement below, but I've lost my reference and am unable to find another one. So, I'm posting this question to see if someone can give a reference, or at least confirm the statement is true (or make a correction).



> 
> **Proposition:** Let $\mathcal{C}$ be a class of schemes with the following properties:
> 
> 
> * $\mathcal{C}$ contains all affine schemes
> * If $X$ is a scheme, $\{ U, V \}$ is an open cover of $X$, and $\mathcal{C}$ contains the three schemes $U$, $V$, and $U \cap V$, then $\mathcal{C}$ contains $X$.
> 
> 
> Then $\mathcal{C}$ contains every quasi-compact quasi-separated scheme. Conversely, the class of quasi-compact quasi-separated schemes has the properties above.
> 
> 
> 



Quasi-compact quasi-separated induction?
This is proposition 3.3.1 and remark 3.3.2 of *Generators and representability of functors in commutative and noncommutative geometry* by A. Bondal, M. van den Bergh.


I originally found this reference via the MO question [The biggest class of schemes which the reduction principle holds](https://mathoverflow.net/q/157199), and I have now found it again by browsing the the Related links to my question, so hurrah for the MO system.


