Hi binu G,
An Op-Amp gain is independent of the rail supply voltages, dual or single. It is the inherent Open-Loop Voltage Gain, Av = Vo/Vi. i.e. it is also the Differential Gain between the Inverting and the Non-Inverting Terminal. Such is a simple Comparator.
Dual Supply is usually used to provide a wider rail-to-rail bipolar voltage swing at the output, just as well as detection at the input, than a Single Supply.
A Single Supply can also provide bipolar voltage swing if the Non-Inverting Terminal is made as the crossing or reference voltage.
When you use a negative feedback path from output to the Inverting Terminal of the Op-Amp, the entire gain is only dependent on the external circuit components(capacitors and resistors). This gain is not the Open-Loop or the Op-Amp Differential Gain. It is the Op-Amp Circuit Gain, i.e. gain based on the external circuit.
External circuit is the feedback circuit and the bias circuit as a whole.
Added after 2 minutes:
To previous post, you can also use the Inverting Terminal as the crossing reference as well, depending what sort of detection you want to make.