In this paper

on basis of the determinant for the metric tensor

together with the identities got via anti-symmetrizing the Kronecker delta symbol and Levi-Civita tensor

we prove one of the properties of the Levi-Civita symbol (tensor)

which shows that two Levi-Civita symbols (tensors) with the contraction for some or total indices can be expressed by a series of Kronecker delta symbols or the generalized Kronecker delta symbol. We further present two important deductions from the property of the Levi-Civita tensor

which are proved in other ways. Besides

we analyze several important properties for the generalized Kronecker delta symbol.