The available evidence suggests that that's much less true than the obvious Economics 101 arguments would suggest. See e.g. https://en.wikipedia.org/wiki/Minimum_wage#Empirical_studies for some pointers.
Obviously if you raise the minimum wage high enough then it's going to hurt employment. But it seems like it has to be quite high before it hurts employment much, and in particular it seems that "everyone who would otherwise have been paid less than the minimum wage just loses their job" is a really inaccurate model.
That is hardly an improvement.