Theorem dfdisjALTV4

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV4 . (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	dfdisjALTV4	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		dfdisjALTV2	⊢ ( Disj 𝑅 ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) )
2		cosscnvssid4	⊢ ( ≀ ◡ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 )
3	2	anbi1i	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )
4	1 3	bitri	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )