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Theorem bj-opelidb 35554
Description: Characterization of the ordered pair elements of the identity relation.

Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than which already appears in the proof. Here for instance this could be the definition I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.)

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

Proof of Theorem bj-opelidb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-id 5529 . . . 4 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
21a1i 11 . . 3 (⊤ → I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
3 eqeq12 2754 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
43adantl 482 . . 3 ((⊤ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥 = 𝑦𝐴 = 𝐵))
52, 4opelopabbv 35545 . 2 (⊤ → (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)))
65mptru 1548 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wtru 1542  wcel 2106  Vcvv 3443  cop 4590  {copab 5165   I cid 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166  df-id 5529
This theorem is referenced by:  bj-opelidb1  35555  bj-opelid  35558
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