Theorem eqvreldisj2

Description: The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 ). (Contributed by Mario Carneiro, 10-Dec-2016) (Revised by Peter Mazsa, 19-Sep-2021)

	Ref	Expression
Assertion	eqvreldisj2	$$⊢EqvRelR\to ElDisj\left(A/R\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj1	$$⊢EqvRelR\to \forall x\in \left(A/R\right)\forall y\in \left(A/R\right)\left(x=y\vee x\cap y=\varnothing \right)$$
2		dfeldisj5	$$⊢ElDisj\left(A/R\right)\leftrightarrow \forall x\in \left(A/R\right)\forall y\in \left(A/R\right)\left(x=y\vee x\cap y=\varnothing \right)$$
3	1 2	sylibr	$$⊢EqvRelR\to ElDisj\left(A/R\right)$$