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Metamath Proof Explorer

Theorem dfdisjALTV5a

Description: Alternate definition of the disjoint relation predicate. Disj R means: different domain generators have disjoint cosets (unless the generators are equal), plus Rel R for relation-typedness. This is the characterization that makes canonicity/uniqueness arguments modular. It is the starting point for the entire " Disj <-> unique representative per block" pipeline that feeds into Disjs , see dfdisjs7 . (Contributed by Peter Mazsa, 3-Feb-2026)

Ref Expression
Assertion dfdisjALTV5a ( Disj 𝑅 ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) ≠ ∅ → 𝑢 = 𝑣 ) ∧ Rel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 dfdisjALTV5 ⊢ ( Disj 𝑅 ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )
2 orcom ⊢ ( ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ↔ ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ∨ 𝑢 = 𝑣 ) )
3 neor ⊢ ( ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ∨ 𝑢 = 𝑣 ) ↔ ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) ≠ ∅ → 𝑢 = 𝑣 ) )
4 2 3 bitri ⊢ ( ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ↔ ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) ≠ ∅ → 𝑢 = 𝑣 ) )
5 4 2ralbii ⊢ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ↔ ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) ≠ ∅ → 𝑢 = 𝑣 ) )
6 1 5 bianbi ⊢ ( Disj 𝑅 ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) ≠ ∅ → 𝑢 = 𝑣 ) ∧ Rel 𝑅 ) )