1st and 2nd laws of thermodynamics

[fabius53]

While refreshing some thermodynamics for a personal interest in renewable energies, I drafted a reasoning that apparently leads from the 1st law to the 2nd law of thermodynamics, so that the latter, at least for ideal gases undergoing reversible processess, would seem to be just a consequence of the former.
Shortly, the reasoning starts from 1st law and ideal gas law, through Mayer's relation, the dependence of internal energy on just temperature, a general expression of the heat exchange $$\delta$$Q, adiabatic slope, Carnot's theorem and finally 2nd law; in detail, it is included here (1stlaw-2ndlaw.pdf).
As far as I remember, 1st law and 2nd law of thermodynamics are basic laws (at least without involving statistical thermodynamics), so I tend to think that there must be some flaw in my reasoning.
Can anyone help me spot any flaws ? Or is the reasoning correct only under very restrictive hypotheses ? Can anyone maybe help me understand more deeply thermodynamics laws ?
I appreciate any contribution

 

[mfb]

You use relations for ideal gases that were derived assuming the second law of thermodynamics in one way or another. It is not surprising that you can extract it again from these relations.