Theorem eldisjs2

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

	Ref	Expression
Assertion	eldisjs2	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		eldisjs	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) )
2		cosselcnvrefrels2	⊢ ( ≀ ◡ 𝑅 ∈ CnvRefRels ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ ≀ ◡ 𝑅 ∈ Rels ) )
3		cosscnvelrels	⊢ ( 𝑅 ∈ Rels → ≀ ◡ 𝑅 ∈ Rels )
4	3	biantrud	⊢ ( 𝑅 ∈ Rels → ( ≀ ◡ 𝑅 ⊆ I ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ ≀ ◡ 𝑅 ∈ Rels ) ) )
5	2 4	bitr4id	⊢ ( 𝑅 ∈ Rels → ( ≀ ◡ 𝑅 ∈ CnvRefRels ↔ ≀ ◡ 𝑅 ⊆ I ) )
6	5	pm5.32ri	⊢ ( ( ≀ ◡ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) )
7	1 6	bitri	⊢ ( 𝑅 ∈ Disjs ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) )