This is called the "bigon criterion". For a discussion, see Section 1.2.4 (and in particular Proposition 1.7) of the "Primer on mapping class groups" by Farb and Margalit.

This is called the "bigon criterion". For a discussion, see Section 1.2.4 (and in particular Proposition 1.7) of the "Primer on mapping class groups" by Farb and Margalit.

The Google search "bigon criterion" also finds various references and lecture notes. For example, here is the top hit:

https://math.stackexchange.com/questions/1646340/proof-of-the-bigon-criterion