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I am beginner in statistical mechanics and found a question asking to show how Fermi-Dirac distribution leads to the explanation of Pauli exclusion principle.

I know how to derive Fermi-Dirac distribution starting from micro canonical ensemble assuming Pauli exclusion principle to be true.

But I don't how to prove Pauli exclusion principle based on Fermi-Dirac distribution expression.

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As you say, you derive the Fermi-Dirac distribution from the Pauli exclusion principle.

An easy way to see the Pauli exclusion principle recovered is in the low temperature limit $T \rightarrow 0$. In this limit, $\left({e^{\frac{E-\mu}{T}}+1}\right)^{-1}$ tends to 1 for $E<\mu$ and 0 for $E>\mu$. This excludes more than 1 electron in any particular energy state.

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I agree with @CameronGibson's answer. Just to mention that the Pauli exclusion principle is derived from the fact that the ket describing a system of multiple fermions is antisymmetric under the exchange of labels. See, for example, Wikipedia - Slater determinant.

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