Theorem bj-opelidb1ALT

Description: Characterization of the couples in _I . (Contributed by BJ, 29-Mar-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-opelidb1ALT	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ I ↔ ( 𝐴 ∈ V ∧ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-br	⊢ ( 𝐴 I 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ I )
2		reli	⊢ Rel I
3	2	brrelex1i	⊢ ( 𝐴 I 𝐵 → 𝐴 ∈ V )
4	1 3	sylbir	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ I → 𝐴 ∈ V )
5		inex1g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∩ 𝐵 ) ∈ V )
6		bj-opelid	⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ V → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ I ↔ 𝐴 = 𝐵 ) )
7	5 6	syl	⊢ ( 𝐴 ∈ V → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ I ↔ 𝐴 = 𝐵 ) )
8	4 7	biadanii	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ I ↔ ( 𝐴 ∈ V ∧ 𝐴 = 𝐵 ) )