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Theorem bj-opelidb1ALT 35985
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
StepHypRef Expression
1 df-br 5148 . . 3 (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2 reli 5824 . . . 4 Rel I
32brrelex1i 5730 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3sylbir 234 . 2 (⟨𝐴, 𝐵⟩ ∈ I → 𝐴 ∈ V)
5 inex1g 5318 . . 3 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
6 bj-opelid 35975 . . 3 ((𝐴𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
75, 6syl 17 . 2 (𝐴 ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
84, 7biadanii 821 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cin 3946  cop 4633   class class class wbr 5147   I cid 5572
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682
This theorem is referenced by: (None)
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