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Theorem bj-elsnb 35345
Description: Biconditional version of elsng 4587. (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 3459 . 2 (𝐴 ∈ {𝐵} → 𝐴 ∈ V)
2 elsng 4587 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2biadanii 819 1 (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  {csn 4573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-sn 4574
This theorem is referenced by: (None)
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