Theorem eqvrelqseqdisj5

Description: Lemma for the Partition-Equivalence Theorem pet2 . (Contributed by Peter Mazsa, 15-Jul-2020) (Revised by Peter Mazsa, 22-Sep-2021)

	Ref	Expression
Assertion	eqvrelqseqdisj5	⊢ ( ( 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		disjimxrn	⊢ ( Disj ( ◡ E ↾ 𝐴 ) → Disj ( 𝑆 ⋉ ( ◡ E ↾ 𝐴 ) ) )
3	1 2	syl	⊢ ( ( EqvRel 𝑅 ∧ ( 𝐵 / 𝑅 ) = 𝐴 ) → Disj ( 𝑆 ⋉ ( ◡ E ↾ 𝐴 ) ) )