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Metamath Proof Explorer

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 ) ) )