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Metamath Proof Explorer

Theorem eqvreldisj3

Description: The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 ). (Contributed by Mario Carneiro, 10-Dec-2016) (Revised by Peter Mazsa, 20-Jun-2019) (Revised by Peter Mazsa, 19-Sep-2021)

		Ref	Expression
	Assertion	eqvreldisj3	⊢ ( EqvRel 𝑅 → Disj ( ^◡ E ↾ ( 𝐴 / 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj2	⊢ ( EqvRel 𝑅 → ElDisj ( 𝐴 / 𝑅 ) )
2		df-eldisj	⊢ ( ElDisj ( 𝐴 / 𝑅 ) ↔ Disj ( ^◡ E ↾ ( 𝐴 / 𝑅 ) ) )
3	1 2	sylib	⊢ ( EqvRel 𝑅 → Disj ( ^◡ E ↾ ( 𝐴 / 𝑅 ) ) )