Definition of a Fiber Bundle

I do not especially like commutation diagrams and I say symbolically the same thing as the Wikipedia definition:
A fiber bundle is (E, B, π, F) where: My φ is the inverse of the φ of the article.

Auxilliary Definition of Atlas

A chart is a map from some non empty open set in B into E.
Question: Is an atlas necessary, not as part of the ‘data’ but as something that must exist in order to be a fiber bundle? Intuitively it seems to me that an atlas could be constructed for any fiber bundle such as defined above. Given an atlas, the above seems to be satisfied; for any x in E x will be in some chart of the atlas and

I see no ready definition of equivalence between fiber bundles.