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Theorem bj-elid3 34531
Description: Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.)
Assertion
Ref Expression
bj-elid3 (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

Proof of Theorem bj-elid3
StepHypRef Expression
1 vex 3483 . 2 𝑥 ∈ V
2 bj-opelidb1 34517 . 2 (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
31, 2mpbiran 708 1 (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2115  Vcvv 3480  cop 4556   I cid 5446
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 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-opab 5115  df-id 5447
This theorem is referenced by: (None)
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