Metamath Proof Explorer


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 𝑅𝑣 ∈ 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 𝑅𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )
3 1 2 bitri ( Disj 𝑅 ↔ ( ∀ 𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )