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 ) )