Metamath Proof Explorer
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 ) ) |