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Metamath Proof Explorer

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	⊢ ( EqvRel 𝑅 → ElDisj ( 𝐴 / 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj1	⊢ ( EqvRel 𝑅 → ∀ 𝑥 ∈ ( 𝐴 / 𝑅 ) ∀ 𝑦 ∈ ( 𝐴 / 𝑅 ) ( 𝑥 = 𝑦 ∨ ( 𝑥 ∩ 𝑦 ) = ∅ ) )
2		dfeldisj5	⊢ ( ElDisj ( 𝐴 / 𝑅 ) ↔ ∀ 𝑥 ∈ ( 𝐴 / 𝑅 ) ∀ 𝑦 ∈ ( 𝐴 / 𝑅 ) ( 𝑥 = 𝑦 ∨ ( 𝑥 ∩ 𝑦 ) = ∅ ) )
3	1 2	sylibr	⊢ ( EqvRel 𝑅 → ElDisj ( 𝐴 / 𝑅 ) )