Majority loser criterion
Either of the Condorcet loser criterion or the mutual majority criterion implies the majority loser criterion. However, the Condorcet criterion does not imply the majority loser criterion (though the Generalized Condorcet criterion does). Neither does the majority criterion imply the majority loser criterion (because the majority is only united in preferring any candidate over the majority loser; they may not necessarily have a single 1st choice candidate that must win).
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The majority loser criterion could be generalized into the mutual majority loser criterion: if a majority prefers any candidates other than those in a given set, then the candidates in the set must not win.