That's absolutely correct -- trichotomy for arbitrary cardinals (any two cardinals are cardinal-size-comparable) requires AC, but SB doesn't require AC. Trichotomy for the cardinal numbers of well-ordered sets (e.g. ordinals) doesn't require AC.
It's a little irrelevant to this thread... but as long as I'm quoting non-proofs that require lots of extra machinery, I'll give my favorite appeal-to-intuition equivalent of choice: the product of non-empty sets is non-empty (any point in the product of a collection of non-empty sets is a choice function on those sets).
It's a little irrelevant to this thread... but as long as I'm quoting non-proofs that require lots of extra machinery, I'll give my favorite appeal-to-intuition equivalent of choice: the product of non-empty sets is non-empty (any point in the product of a collection of non-empty sets is a choice function on those sets).