I'm studying for my thermodynamics exam and I came across something which really confuses me.
An infinitesimal change in entropy $ dS_{sys}$ of a system at temperature $T_{sys}$ during a reversible transformation, where $\delta Q_{rev}$ is defined as the heat going in/out the system is given by: $$ dS_{sys} = \frac{\delta Q_{rev}}{T_{sys}} $$
However, there is a statement in my book claiming that: $ dS_{sys} > \frac{\delta Q_{rev}}{T_{surr}} $
My confusion is the following: If an infinitesimal change in entropy of a system at temperature $T_{sys}$ is defined as above, how can the statement $ dS_{sys} > \frac{\delta Q_{rev}}{T_{surr}} $ be true? In order to calculate the change in entropy the path must be reversible, meaning that the temperature of the system is equal to the temperature of the surroundings, i.e. $ T_{sys} = T_{surr} $ otherwise the path isn't reversible. The statement clearly doesn't hold if my reasoning is correct.
Can someone clarify this to me because I'm really struggling with this.