Antireflexive and Transitive Relation is Antisymmetric
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Theorem
Let $\RR \subseteq S \times S$ be a relation which is not null.
Let $\RR$ be antireflexive and transitive.
Then $\RR$ is also antisymmetric.
Proof
Let $\RR \subseteq S \times S$ be antireflexive and transitive.
From Antireflexive and Transitive Relation is Asymmetric it follows that $\RR$ is asymmetric.
The result follows from Asymmetric Relation is Antisymmetric.
$\blacksquare$
Also see
If $\RR = \O$ then Null Relation is Antireflexive, Symmetric and Transitive applies instead.
Sources
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $4$