Theorem disjim

Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 , cf. eldisjim . (Contributed by Peter Mazsa, 3-May-2019) (Revised by Peter Mazsa, 17-Sep-2021)

	Ref	Expression
Assertion	disjim	⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		dfdisjALTV4	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑦 ∃* 𝑢 𝑢 𝑅 𝑦 ∧ Rel 𝑅 ) )
2	1	simplbi	⊢ ( Disj 𝑅 → ∀ 𝑦 ∃* 𝑢 𝑢 𝑅 𝑦 )
3		trcoss	⊢ ( ∀ 𝑦 ∃* 𝑢 𝑢 𝑅 𝑦 → ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ≀ 𝑅 𝑦 ∧ 𝑦 ≀ 𝑅 𝑧 ) → 𝑥 ≀ 𝑅 𝑧 ) )
4	2 3	syl	⊢ ( Disj 𝑅 → ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ≀ 𝑅 𝑦 ∧ 𝑦 ≀ 𝑅 𝑧 ) → 𝑥 ≀ 𝑅 𝑧 ) )
5		eqvrelcoss3	⊢ ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ≀ 𝑅 𝑦 ∧ 𝑦 ≀ 𝑅 𝑧 ) → 𝑥 ≀ 𝑅 𝑧 ) )
6	4 5	sylibr	⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅 )