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Theorem bj-elsnb 34450
Description: Biconditional version of elsng 4564. (Contributed by BJ, 18-Nov-2023.)
Assertion
Ref Expression
bj-elsnb (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-elsnb
StepHypRef Expression
1 elex 3498 . 2 (𝐴 ∈ {𝐵} → 𝐴 ∈ V)
2 elsng 4564 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2biadanii 821 1 (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550
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-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sn 4551
This theorem is referenced by: (None)
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