  Chemical bimodality of the Milky Way (MW) disc stars constitutes one of the
most remarkable properties of MW. The cold accretion theory for the
cosmological gas accretion provides one viable explanation to this phenomenon.
In this scenario, the rapid cold-mode accretion in the early epoch creates the
first generation stars relatively rich in $\alpha$-elements(O,Mg,Si,S,Ca,etc)
and later cooling flow produces iron-rich second generation stars, creating the
bimodality in the [$\alpha$/Fe] ratio. We employ a cosmologically motivated
chemical evolution model for disc galaxies to elucidate the role played by type
Ia supernovae (SNIa), which serve as the major source of iron, in the creation
of the bimodality. To this end, we divide SNIa into two groups, those formed
from the 1st generation stars (the first SNIa) and those formed from the 2nd
generation stars (the second SNIa). The model with the first SNIa suppressed
during the {\it second} star formation stage produces stars having high
[$\alpha$/Fe] in the early phase of this stage, whereas the model which
prohibits the second SNIa produces high [$\alpha$/Fe] stars in the late phase.
Both models fail to create a well-defined bimodality. We thus conclude that the
cooperation of the first and the second SNIa plays a crucial role in creating
the bimodality by maintaining rich iron content in the interstellar gas
throughout the second star formation stage.
