Theorem eldisjs

Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021)

	Ref	Expression
Assertion	eldisjs	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		dfdisjs	⊢ Disjs = { 𝑟 ∈ Rels ∣ ≀ ◡ 𝑟 ∈ CnvRefRels }
2		cnveq	⊢ ( 𝑟 = 𝑅 → ◡ 𝑟 = ◡ 𝑅 )
3	2	cosseqd	⊢ ( 𝑟 = 𝑅 → ≀ ◡ 𝑟 = ≀ ◡ 𝑅 )
4	3	eleq1d	⊢ ( 𝑟 = 𝑅 → ( ≀ ◡ 𝑟 ∈ CnvRefRels ↔ ≀ ◡ 𝑅 ∈ CnvRefRels ) )
5	1 4	rabeqel	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) )