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Theorem bj-elsnb 37044
Description: Biconditional version of elsng 4645. (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 3499 . 2 (𝐴 ∈ {𝐵} → 𝐴 ∈ V)
2 elsng 4645 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2biadanii 822 1 (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sn 4632
This theorem is referenced by: (None)
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