We select a sample of 1437 active galactic nuclei (AGN) from the catalog of the Sloan Digital Sky Survey galaxy properties from the Portsmouth group by detection of the high-ionization [NeV]3426{AA} emission line. We compare the fluxes of [NeIII]3869{AA}, [OIII]5007{AA}, [OII]3726,3728{AA}, and [OI]6300{AA} to that of [NeV]. All four lines show a strong linear correlation with [NeV], although lines from ions with lower ionization potentials have a lower correlation coefficient. We investigate the use of two forbidden line ratio diagnostic diagrams that do not rely on H{alpha} in order to classify high-redshift galaxies. These use the [NeIII]/[OII] line ratio plotted against [OIII]/[OI] and [OIII]/[OII], respectively. We use photoionization modeling to characterize the behavior of the narrow-line region in AGN and star-forming regions and test the validity of our diagnostic diagrams. We also use a luminosity cutoff of logL[OIII][erg/s]=42, which lowers the contamination of the AGN region by star-forming galaxies down to 10% but does not remove green pea and purple grape galaxies from the AGN region. We also investigate the OHNO diagram, which uses [NeIII]/[OII] plotted against [OIII]/H{beta}. Using our new diagnostic diagrams, we are able to reliably classify AGN up to a redshift of z<=1.06 and add more than 822 new AGN to the [NeV]-selected AGN sample.