"I am not a teacher, but a student."
Translation:Én nem tanár vagyok, hanem diák.
You MUST include "vagyok" because it is the verb. As you probably know, "van" is one case where the verb is often omitted. That is because "van" is 3rd person. "Vagyok" has to be included, since it is 1st person.
"Én" on the other hand is just a pronoun, which can be left out. It is only used for emphasis, like when comparing I vs. you, or something like that.