Fix a language $\mathcal{L}:=\{=\}$, and let $T$ be the $\mathcal{L}$-theory of singleton collections. Namely the theory axiomatized by the $\mathcal{L}$-sentence $$\sigma:=\exists x((x=x)\wedge(\forall y(y=y\implies y=x))).$$
From what I can tell, there are no meaningful consequences of this theory. Aside from the size of the collection being restricted, it doesn't seem to me that I can conclude anything else universal about any potential model of $T$ that is meaningful. Hence, my question:
Are there any universal $\mathcal{L}$-sentences in $T$, aside from the obvious ones like $\forall x(x=x)$ or $\forall x\forall y(\neg y=x\implies \neg x=y)$?
(By universal $\mathcal{L}$-sentences, I'm talking about sentences of the form $\forall x\phi(x)$.)
Context: I'm trying to see if there is anything restricting a two element collection from modelling the universal theory $T_{\forall}$ of $T$, where $T_{\forall}$ is the set of $\mathcal{L}$-sentences in $T$ which are universal and their consequences.