Given the theory (set of FOL formulas)
$$ T = \{\forall x_1 \neg Q(x_1,x_1), \forall x_1 \exists x_2 Q(x_1,x_2), \forall x_1 \forall x_2 \forall x_3 (Q(x_1,x_2) \land Q(x_2,x_3)) \to Q(x_1,x_3)\} $$
Show that every model for $T$ is infinite.
I tried to assume that there exists a finite model, but I could not see the contradiction. Any suggestion?