is there a way to prove that a number is an integer?
You are asking a foundational (set theory) question.
This theory is taught in any well constructed Foundation of Mathematics course.
You can get some ideas from this webpage.
Writing x < 0 does not make x negative or change x in any way. Rather, for each value of x, the formula x < 0 is true or false. As far as notation goes, you can write $\displaystyle x\in\mathbb{Z}$ or just write "x is an integer"; this statement would be true iff x is an integer. If you need to express the property of being an integer using other concepts, then the answer depends on what you are allowed to use. For example, if you are working with rational numbers and are allowed to use the ceiling function $\displaystyle \lceil\cdot\rceil$, then x is an integer iff $\displaystyle x = \lceil x\rceil$. In the context of set theory see Plato's response.