Theorem dfdisjALTV5

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	dfdisjALTV5	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		dfdisjALTV2	⊢ ( Disj 𝑅 ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) )
2		cosscnvssid5	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )
3	1 2	bitri	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )