Two-dimensional (2D) electrides are a new concept material in which anionic electrons are confined in the interlayer space between positively charged layers. We have performed angle-resolved photoemission spectroscopy measurements on Y$_2$C, which is a possible 2D electride, in order to verify the formation of 2D electride states in Y$_2$C. We clearly observe the existence of semimetallic electride bands near the Fermi level, as predicted by ${ab}$ ${initio}$ calculations, conclusively demonstrating that Y$_2$C is a quasi-2D electride with electride bands derived from interlayer anionic electrons.