It does look somewhat confusing, because the vertical bar |, which in this case means And, looks similar to the double bar ||, which means Or. Also, the name "Alternatives" usually means "either or", not And. However the documentation of Element states that the three following inputs are equivalent:

Element[x | y, Integers]
Element[x, Integers] && Element[y, Integers]
Element[{x, y}, Integers]

and they are different from

Element[x, Integers] || Element[y, Integers]

It is a design choice that probably has its rationale.

Yeah, I said so too; quite unexpected output to be honest. But I see now that it is indeed returned by Reduce sometimes.

Please indent your code using tab in front of each line such that it is displayed correctly.

What are eq2...eq6 ? I can't reproduce it without all code.

Try minimizing your problem.

Please submit the entire code (and/or entire input), I have a feeling there is an error in it. Normally you expect || in the output (Or). A single | is Alternatives in Wolfram Language but is only used for pattern matching as far as I know.

It means that all the C[i] are integers. See also the documentation.