A density functional formalism is applied to investigate the wetting behavior of Ne confined in slits composed of two parallel solid identical alkaline walls with increasing attractive strength leading to a variety of wetting situations. The study is performed over the complete range of temperature spanned from the triple point T_t up to the critical one T_c of Ne. Attention is paid to the slit's width. It was found that in the case of weaker substrates for temperatures below a certain critical T_sb the density profiles corresponding to the lowest free energy are asymmetric, i.e. exhibit a spontaneous breaking of symmetry. For T > T_sb the phenomenon of symmetry breaking disappears leading to a first-order phase transition.