Theorem eldisjs4

Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	eldisjs4	⊢ ( 𝑅 ∈ Disjs ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		eldisjs2	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) )
2		cosscnvssid4	⊢ ( ≀ ◡ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 )
3	2	anbi1i	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ 𝑅 ∈ Rels ) )
4	1 3	bitri	⊢ ( 𝑅 ∈ Disjs ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ 𝑅 ∈ Rels ) )