Theorem eqvrelqseqdisj4

Description: Lemma for petincnvepres2 . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	eqvrelqseqdisj4	⊢ ( ( EqvRel 𝑅 ∧ ( 𝐵 / 𝑅 ) = 𝐴 ) → Disj ( 𝑆 ∩ ( ◡ E ↾ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj3	⊢ ( ( EqvRel 𝑅 ∧ ( 𝐵 / 𝑅 ) = 𝐴 ) → Disj ( ◡ E ↾ 𝐴 ) )
2		disjimin	⊢ ( Disj ( ◡ E ↾ 𝐴 ) → Disj ( 𝑆 ∩ ( ◡ E ↾ 𝐴 ) ) )
3	1 2	syl	⊢ ( ( EqvRel 𝑅 ∧ ( 𝐵 / 𝑅 ) = 𝐴 ) → Disj ( 𝑆 ∩ ( ◡ E ↾ 𝐴 ) ) )