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Is there a good phrase or word that means the opposite of "coexist"?

The phrase I am using it in is similar to "these two ideas need not (be) __."

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Mutually exclusive

Two things that can't happen at the same time.

http://en.wikipedia.org/wiki/Mutually_exclusive_events

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"These two (noun)s need not _____"

Some ideas:

  1. Disagree
  2. Differ
  3. Conflict
  4. Contradict
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Depends on the context but in many instances the opposite of coexistence is persecution.

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    Persecution isn't suitable for the context given in the question, grammatically or semantically. – Bradd Szonye Apr 23 '13 at 21:22
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    Also, exist derives from Latin existĕre, which you will note is a Latin conjugation other than the 1st (-are) conjugation — and therefore does not take -ance in English but rather -ence. These things matter. – tchrist♦ Apr 23 '13 at 21:25
  • I didn't read the phrase. Yes, given the context, "persecution" is not a valid answer. – Jason Powell Hundley Apr 24 '13 at 20:14
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"disparate" comes to mind. Fundamentally distinct or different in kind; entirely dissimilar:

"diametrically opposed" also comes to mind.

how about "at odds?" or "Immiscible".

or "irreconcilable"?

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