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Theorem bj-elsn12g 35463
Description: Join of elsng 4598 and elsn2g 4622. (Contributed by BJ, 18-Nov-2023.)
Assertion
Ref Expression
bj-elsn12g ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem bj-elsn12g
StepHypRef Expression
1 elsng 4598 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 elsn2g 4622 . 2 (𝐵𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2jaoi 855 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845   = wceq 1541  wcel 2106  {csn 4584
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-sn 4585
This theorem is referenced by: (None)
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