Increment letter instead of Offer Letter

Hi grapevine folks,

I had joined a startup in jan as an intern and had received joining and offer letter.

Now in july I have become full time and have received an increment letter and not an offer letter. Does it create any problem in showing my work ex.

And since it is a startup I dont receive any payslip and directly receive money in my bank account.

Can I use my bank account statement instead of payslips to prove my work ex.

I am asking this with the view of proving my work ex in a b-school.