Question about my understanding of Clausius inequality and changes in entropy

So, I know that $\oint \frac{\delta Q}{T}\le0$ , that's the inequality and it's equal to 0 for reversible and less than 0 for irreversible. But at the same time I know $\Delta S=\oint \frac{\delta Q_{rev}}{T_{surr}}$ and $\frac{\delta Q_{rev}}{T}\ge \frac{\delta Q}{T}$ coming from the Planck-Kelvin statement of the second law of thermodynamics. But doesn't this just mean that $\Delta S \ge \oint \frac{\delta Q}{T} \le0$ , which looks kinda useless. It means that entropy might increase, might still be 0 or might even still decrease as long as the cyclic integral is even lower. How does this show that entropy always increases in an irreversible process? Or does it show that it can decrease as long as the net increase is greater in the surroundings? And if so, how do we prove (again, mathematically) that that is the case and entropy isn't just decreasing in the whole Universe? Thanks for the help in advance!