Let $S$ be a set, $\mathcal{R}$ be a relation on $S^{2}$ such that $(\forall x\in S\mid\neg(x\ \mathcal{R}\ x)\ )$, and $\mathbf{I}_S$ the identity relation on $S$.
At the end of a proof, I jump from $$\neg(\exists x,y\in S\mid\langle x,y\rangle\in\mathbf{I}_S\cap\mathcal{R})$$ to $$(\mathbf{I}_S\cap \mathcal{R}=\emptyset).$$ In your opinion, is that too hasty, considering that the teacher did not give a formal definition of the empty set, only saying it is a set empty of any element?