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Theorem bj-opelidb1 34463
Description: Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 34462 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)
Assertion
Ref Expression
bj-opelidb1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-opelidb1
StepHypRef Expression
1 an32 645 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V))
2 bj-opelidb 34462 . 2 (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
3 eleq1 2899 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
43biimpac 482 . . 3 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
54pm4.71i 563 . 2 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V))
61, 2, 53bitr4i 306 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  cop 4546   I cid 5432
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-opab 5102  df-id 5433
This theorem is referenced by:  bj-idres  34470  bj-elid3  34477  bj-eldiag2  34487
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