Theorem eldisjs3

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

	Ref	Expression
Assertion	eldisjs3	⊢ ( 𝑅 ∈ Disjs ↔ ( ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) ∧ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		eldisjs2	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) )
2		cosscnvssid3	⊢ ( ≀ ◡ 𝑅 ⊆ I ↔ ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) )
3	2	anbi1i	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ ( ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) ∧ 𝑅 ∈ Rels ) )
4	1 3	bitri	⊢ ( 𝑅 ∈ Disjs ↔ ( ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) ∧ 𝑅 ∈ Rels ) )