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Theorem bj-elsn12g 37061
Description: Join of elsng 4640 and elsn2g 4664. (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 4640 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 elsn2g 4664 . 2 (𝐵𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2jaoi 858 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 848   = wceq 1540  wcel 2108  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sn 4627
This theorem is referenced by: (None)
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