In Schroeder's "An Introduction to Thermal Physics" in section 5.2 (page 161), Schroeder considers the case of a system that is in thermal contact with a reservoir that is at a constant temperature $T_\text{R}$. I'll quote the relevant section:
The total entropy of the universe can be written as $S_\text{total}=S+S_\text R$, where a subscript $\text R$ indicates a property of the reservoir, while a quantity without a subscript refers to the system alone... let's consider a small change in the total entropy: $$\text dS_\text{total}=\text dS+\text dS_\text{R}\tag{5.26}\label{5.26}$$ I would like to write this quantity entirely in terms of system variables. To do so, I'll apply the thermodynamic identity, in the form $$\text dS=\frac1T\ \text dU+\frac PT\ \text dV-\frac\mu T\ \text dN\tag{5.27}\label{5.27}$$ to the reservoir. First, I'll assume that $V$ and $N$ for the reservoir are fixed - only energy travels in and out of the system. Then $\text dS_\text R=\text dU_\text R/T$, so equation \eqref{5.26} can be written $$\text dS_\text{total}=\text dS+\frac1T_\text R\text dU_\text R\tag{5.28}\label{5.28}$$ But the temperature of the reservoir is the same as the temperature of the system, while the change $\text dU_\text R$ in the reservoir's energy is minus the change $\text dU$ in the system's energy. Therefore, $$\text dS_\text{total}=\text dS-\frac1T\text dU=-\frac1T(\text dU-T\ \text dS)=-\frac1T\ \text dF\tag{5.29}\label{5.29}$$ Aha! Under these conditions (fixed $T$, $V$, and $N$), an increase in the total entropy of the universe is the same thing as a decrease in the Helmholtz free energy of the system. So we can forget about the reservoir, and just remember that the system will do whatever it can to minimize its Helmholtz free energy.
My question is, if $\text dV$ and $\text dN$ are also $0$ for the system, then shouldn't the thermodynamic identity for the system then be $\text dS=\text dU/T$, leading to $\text dS-\text dU/T$ in equation $\eqref{5.29}$ being equal to $0$?