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Theorem bj-elid3 37149
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 3481 . 2 𝑥 ∈ V
2 bj-opelidb1 37135 . 2 (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
31, 2mpbiran 709 1 (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  wcel 2105  Vcvv 3477  cop 4636   I cid 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5210  df-id 5582
This theorem is referenced by: (None)
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