thermodynamics - How does Fermi jump to this conclusion in Clausius inequality?

In his $Thermodynamics$, Fermi proves beautifully the following (rephrased):

For a system undergoing a cyclic process, $$\oint {\delta Q\over T}\leq 0,$$ and for a reversible cyclic process, it is an equality.

Then he states the following without any proof:

”... and $\oint {\delta Q\over T}= 0$ which is valid only for reversible cycles.”

Question: How does he conclude the converse of the implication as stated above? In other words, how to prove that the Clausius integral evaluating to zero implies the cycle to be reversible? (Without introducing the concept of entropy.)

Edit: Well, you may use entropy, if you can’t prove without it.

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It is mentioned on page $48$ of his classy $Thermodynamics$.