I'm looking for an expression or idiom implying two things contradicting each other.
the original sentence that I want to use this in, translates to something like this:
hatred and self-building contradict each other
or
hatred and self-building don't go well together
though I'm looking for something stronger than "don't go well together" which would imply contradiction.
thanks in advance.
The simplest idiomatic phrase may be at odds—although a number of others may be more suitable to particular situations. Here is the entry for at odds in Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013):
at odds In disagreement, opposed. For example, It is only natural for the young and the old to be at odds over money matters. This idiom uses odds in the sense of "a condition of being unequal or different," and transfers it to a difference of opinion, or quarrel. {Late 1500s}
Ammer suggests that the phrase applies primarily to people and to quarreling, but I have heard the expression used figuratively to refer to things that incompatible, such as philosophical or political positions, even if (as happens more often than you might think) they are held by a single person who is unaware of the conflict in their tendencies or logical consequences.
Another idiom that may be even more suitable to the situation you describe is at cross purposes. Here is Ammer's entry for that idiom:
at cross purposes With aims or goals that conflict or interfere with one another, as in I'm afraid the two departments are working at cross purposes. This idiom, first recorded in 1688, may have begun as a 17th-centuryparlor game called "cross-purposes," in which a series of subjects (or questions) were divided from their explanations (or answers) and distributed around the room. Players then created absurdities by by combining a subject taken from one person with an explanation taken from another.
Other potentially relevant idioms, depending on the intensity of the conflict or incompatibility, include at loggerheads (which Ammer defines as "engaged in a a quarrel or dispute"), and at daggers drawn (which Ammer defines as "about to or ready to fight"). Both of these idioms, however, unlike at odds and at cross-purposes, do seem to be generally restricted to disputes between people.
The answer above is top-notch. There are however a couple of suggestions I'd like to put forward that might work in certain contexts.
"Go against the grain"
Cambridge: If something goes against the grain, you would not usually do it because it would be unusual
But more to the point I would suggest "Fly in the face/ teeth of".
Collins: If an action or belief flies in the face of accepted ideas or rules, it seems to completely oppose or contradict them.
Or simply "go against (something)"
