Equivalence relation - help needed with proving

Question

let E be an Equivalence relation on A (non-empty group) is the next argument true or not, justify your answer:

$\exists x~(\forall y~(xEy)\lor \exists y~\forall z~(zEx\lor zEy)) \to \forall x~\exists y~\exists z((x\neq y)\land (x\neq z)\land (xEy\lor xEz))$

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Let $A = \{1,2,3\}$ and $E = \operatorname{Id}a$

Is that a good counter-example?

Answer 1

The formula is not valid: Let $A$ have only one element.

Your example doesn't satisfy the condition of the implication (has more than two equivalence classes), hence it's not a counterexample.