The law of trichotomy holds for cardinal numbers, so that any two sets are either equinumerous, or one has a strictly smaller cardinality than the other.
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In Zermelo Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice.
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Already in the 1879 " Begriffsschrift " important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic.
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