Theorem bj-opelidb1ALT 37622

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 5100	. . 3 ⊢ (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2		reli 5797	. . . 4 ⊢ Rel I
3	2	brrelex1i 5701	. . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V)
4	1, 3	sylbir 237	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I → 𝐴 ∈ V)
5		inex1g 5274	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
6		bj-opelid 37612	. . 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 831	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903  ⟨cop 4587   class class class wbr 5099   I cid 5539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652

This theorem is referenced by: (None)