Theorem bj-opelidb1ALT 35337

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

Assertion

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

Proof of Theorem bj-opelidb1ALT

Step	Hyp	Ref	Expression
1		df-br 5075	. . 3 ⊢ (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2		reli 5736	. . . 4 ⊢ Rel I
3	2	brrelex1i 5643	. . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
4	1, 3	sylbir 234	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I → 𝐴 ∈ V)
5		inex1g 5243	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
6		bj-opelid 35327	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7	5, 6	syl 17	. 2 ⊢ (𝐴 ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
8	4, 7	biadanii 819	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886  ⟨cop 4567   class class class wbr 5074   I cid 5488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596

This theorem is referenced by: (None)