Lemma. (Transitivity) "$\leq_p$" is a transitive relation on languages, i.e., if $L_1 \leq_p L_2$ and $L_2 \leq_p L_3$, then $L_1 \leq_p L_3$.
Proof. By definition, there are poly-time functions $f$ and $g$ such that $x \in L_1 \Leftrightarrow f(x) \in L_2$ and $y \in L_2 \Leftrightarrow g(y) \in L_3$, thus $x \in L_1 \Leftrightarrow f(x) \in L_2 \Leftrightarrow g(f(x)) \in L_3$. Obviously, $g(f(\cdot))$ is poly-time (since $|f(x)|$ is polynomial in $|x|$).
This lemma proves that "$\leq_p$" is transitive, but how would I prove that it is not antisymmetric?