The Schroeder-Bernstein theorem relies on the trichotomy law, and therefore on it. Tychonoff's theorem, which states that the product space of compact spaces is itself compact also uses it, as does the proof that every vector space has a linearly independent spanning set. It is consistent with von-Neumann-Bernays-Godel set 
The Schroeder-Bernstein theorem relies on the trichotomy law, and therefore on it. Tychonoff's theorem, which states that the product space of compact spaces is itself compact also uses it, as does the proof that every vector space has a linearly independent spanning set. It is consistent with von-Neumann-Bernays-Godel set theory, and both it and its negation are consistent with, and in fact independent from, the Zermelo-Frankel axioms of set theory. Yielding such results as the Banach-Tarsky paradox and originally stating that given any set of mutually exclusive nonempty sets there exists a set that contains exactly one element in common with the nonempty sets, for 10 points, identify this mathematical statement which is equivalent to Zorn's Lemma and the well-ordering principle.
The Schroeder-Bernstein theorem relies on the trichotomy law, and therefore on it.
{axiom of choice} [accept {Zorn’s Lemma} or {well-ordering} before “Zermelo-Frankel” is read]