There is not much to prove there; opposite numbers, by which I take you mean "additive inverse", are defined so that their sum equals zero.
Zero
The answer depends on what is meant by "their opposites". If you mean additive opposites then the set is of all non-zero integers.
A zero pair
It is zero and that is simply because that is how additive opposites are defined.
Zero
Their total is zero.
Two integers are considered opposites if they are equal in absolute value but have different signs. For example, 5 and -5 are opposites because they are the same distance from zero on the number line, but one is positive and the other is negative. When added together, opposites always yield a sum of zero, illustrating their relationship.
Because by definition, there sum will always be zero. Because the definition of opposites are additive inverses.
opposites
additive inverse property
These are the integers.
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