I am reading Concepts in Thermal Physics (2nd ed) by S. J. Blundell and K. M. Blundell. In section 16.5 (Thermodynamic potentials - Constraints) there is an explanation of the role of the Helmholtz free energy in an isothermic process. The end result is that $đW \geq dF$ (the work done on the system is greater than the increase in the Helmholtz function).
The derivation begins (paraphrased): Consider a system in contact with a reservoir at temperature $T$. If heat $đQ$ enters the system then the entropy change of the surroundings is $dS_0 = -đQ/T$ and the entropy change $dS$ of the system has to be such that $0 \leq dS_\mathrm{universe} = dS + dS_0$, which gives $dS \geq đQ/T$, which is a statement of the Clausius inequality. The derivation goes on for a few steps, but my problem is with the beginning:
If the process is irreversible ($dS_\mathrm{universe} > 0$), how can it be that $dS_0 = -đQ/T$? The first law is $dU = đQ + đW = TdS - pdV$, but the book makes it very clear that $đQ = TdS$ and $đW = -pdV$ only for reversible processes (for irreversible processes $đQ < TdS$ and $đW > -pdV$). Is there some hidden assumption made, or am I confused about something elementary? (the latter seems far more probable...)