Prove the relation $R = \{(x,y)\in \mathbb{R} \times \mathbb{R}: \text{ } |x|< |y|\text{ or $x=y$} \}$ is antisymmetric.
Proof:
Suppose $ x R y$ and $ yRx $. Then $|x|<|y|$ or $x=y$. Moreover, $|x|>|y|$ or $y=x$.
Case $|x|<|y|$:
Then because $yRx$, $x=y$.
Case $x=y$: Done.
In either case we showed $x=y$; hence, $R$ is antisymmetric.
Is the above proof correct?