Explaining the conditionals

First we have the zero conditional, the structure is:

If+ present simple+present simple
example: If I arrive to class late, my teacher gets angry.

-First conditional:
If+ present simple+ will/won´t
example: if I have a dog, I will have to care of him.

-Second conditional:
If+ past simple+ would/wouldn´t
example: If she lost her bag, she would buy a new one.

-Third conditional:
If+ past perfect+ would have+ past participle
example: If he had been working, he would have saved money.

These are the the zero, first, second and thir conditionals that we have been learning.