The various large cardinal axioms are usually described in terms of some roughly linear hierarchy of varying consistency strengths. However, some cardinal axioms potentially contradict one another. As an example, asserting the existence of unboundedly many strong limit cardinals implies AC. However, asserting the existence of Reinhardt cardinals or Berkeley cardinals implies the negation of AC (all in ZF). This seems to imply different mutually exclusive hierarchies of large cardinal axioms in ZF.

Are there other mutually exclusive LCA's in ZFC? If so, what are some of the exact statements that they conflict on? If not, why wouldn't we expect this in ZFC when it seems to be the case in ZF?