Equivalence of Definitions of Antisymmetric Relation (Class Theory)

Definition 1 implies Definition 2
Let $\RR$ be a relation which fulfils the condition:


 * $\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$

Let $\tuple {x, y} \in \RR$ such that $x \ne y$.

that $\tuple {y, x} \in \RR$.

Then $\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR$.

Then this implies that $x = y$.

From this contradiction it is concluded that $\tuple {y, x} \notin \RR$.

It follows that the condition:
 * $\tuple {x, y} \in \RR \land x \ne y \implies \tuple {y, x} \notin \RR$

holds for $\RR$.

Definition 2 implies Definition 1
Let $\RR$ be a relation which fulfils the condition:


 * $\tuple {x, y} \in \RR \land x \ne y \implies \tuple {y, x} \notin \RR$

Let $\tuple {x, y} \in \RR$ such that $\tuple {y, x} \in \RR$ also.

that $x \ne y$.

Then this implies that $\tuple {y, x} \notin \RR$.

From this contradiction it is concluded that $\tuple {y, x} \notin \RR$.

It follows that the condition:
 * $\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$

holds for $\RR$.