Problem
Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes.
Current Solution
First, the axioms for equivalence class:
\begin{alignat*}{3} &\forall x &&E(x,x),\\ &\forall x \forall y &&(E(x,y) \rightarrow E(y,x)),\\ &\forall x \forall y \forall z &&(E(x,y) \wedge E(y,z) \rightarrow E(x,z))).\\ \end{alignat*}
Then to express infinitely many equivalence classes, we add infinitely many senteneces $\phi_n ~(n \ge 2$): $$\phi_n = \exists x_1 \ldots \exists x_n ~ \bigwedge_{i < j \le n} \neg E(x_i,x_j).$$
That is we have sentences $\phi_2,\phi_3,\ldots$. Finally, we add infinitely many sentences $\psi_n ~(n \ge 1)$ $$\psi_n = \forall x \exists x_1 \ldots \exists x_n ~ \bigwedge_{i < j \le n} x_i \neq x_j \wedge \bigwedge_{i=1}^n E(x,x_j)$$
to axiomize each class is infinite.
Problem
I am not confident that $\{\phi_n\}$ and $\{\psi_n\}$ are not contradicting each other