A search for the heaviest isotopes of fluorine, neon, and sodium was conducted by fragmentation of an intense ^{48}Ca beam at 345  MeV/nucleon with a 20-mm-thick beryllium target and identification of isotopes in the large-acceptance separator BigRIPS at the RIKEN Radioactive Isotope Beam Factory. No events were observed for ^{32,33}F, ^{35,36}Ne, and ^{38}Na and only one event for ^{39}Na after extensive running. Comparison with predicted yields excludes the existence of bound states of these unobserved isotopes with high confidence levels. The present work indicates that ^{31}F and ^{34}Ne are the heaviest bound isotopes of fluorine and neon, respectively. The neutron dripline has thus been experimentally confirmed up to neon for the first time since ^{24}O was confirmed to be the dripline nucleus nearly 20 years ago. These data provide new keys to understanding the nuclear stability at extremely neutron-rich conditions.