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Theorem bj-elid3 34351
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 3495 . 2 𝑥 ∈ V
2 bj-opelidb1 34337 . 2 (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
31, 2mpbiran 705 1 (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  Vcvv 3492  cop 4563   I cid 5452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-id 5453
This theorem is referenced by: (None)
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