We know the results, add water to ice. Ice melts and water cools. How do we PROVE that it is 100% Hot to Cold? The molecules in a hot object are moving faster and will facilitate the transfer of heat to the colder medium. BUT, its molecules are also moving. How do we PROVE it is 100% in one direction and Not 95%/5% or some other ratio until equilibrium is achieved?
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The laws of thermodynamics possess a probabilistic nature: they are expected to hold on average, but there is nothing to preclude their temporary violation when we go beyond the thermodynamic limit. As thermodynamics typically involves exceptionally large systems, the law of large numbers tells us that the likelihood of observing significant deviations from the average value essentially becomes negligible.
If you are interested in an insightful demonstration of Clausius's statement, take a look at to Section II, titled 'Why Entropy Production Matters,' in the paper Irreversible Entropy Production, from Quantum to Classical. At the end of Subsection A, the authors present a compelling argument showing that this statement is valid based on considerations of entropy.
Alex
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