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Metamath Proof Explorer

Theorem eqvreldisj4

Description: Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020) (Revised by Peter Mazsa, 31-Dec-2021)

		Ref	Expression
	Assertion	eqvreldisj4	⊢ ( EqvRel 𝑅 → Disj ( 𝑆 ∩ ( ^◡ E ↾ ( 𝐵 / 𝑅 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj3	⊢ ( EqvRel 𝑅 → Disj ( ^◡ E ↾ ( 𝐵 / 𝑅 ) ) )
2		disjimin	⊢ ( Disj ( ^◡ E ↾ ( 𝐵 / 𝑅 ) ) → Disj ( 𝑆 ∩ ( ^◡ E ↾ ( 𝐵 / 𝑅 ) ) ) )
3	1 2	syl	⊢ ( EqvRel 𝑅 → Disj ( 𝑆 ∩ ( ^◡ E ↾ ( 𝐵 / 𝑅 ) ) ) )