Oh, but these are English!
You wrote "двуязычие не редкого явления" (never mind that you originally used the dative ending). The correction was "двуязычие - не редкое явление".
In this expression you say "A is not (does not equal) B". Such expressions use "не" in Russian, and both A and B are in nominative case. This is your "This is not a rare phenomenon".
In "There are no books" you make no comparison, but you deny the existence or presence of books. Russian does this with нет and genitive: нет книг. In Russian you use negation and genitive to express that something is not there or that something is not being done or hasn't been done: что ты делал? (Я делал) ничего особенного. But for A is not B you simply use nominative.
Furthermore you should note the dash in the correction which is used in Russian where a form of "быть" (which used to exist but no longer does) is dropped. To make matters worse, as soon as you do use a form of быть B turns instrumental: он - (не) хороший учитель : он (не) был хорошим учителем.
Only if you say that there is or was no good teacher in general, genitive enters the picture: нет хорошего учителя : не было хорошего учителя. Note было, it was, the verb does not agree grammatically to the noun phrase which follows, as it would in some other languages (frex German), but to a non-disclosed neutral grammatical subject.
Disclaimer: while I am pretty sure this is correct I am no native speaker of Russian or English.
Disclaimer disclaimer: As no cries of outrage have been heard yet and I did receive a reputation comment on this by Lampada this post seems to be not too far off the mark.