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Theorem bj-elsn12g 37043
Description: Join of elsng 4645 and elsn2g 4669. (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 4645 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 elsn2g 4669 . 2 (𝐵𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2jaoi 857 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847   = wceq 1537  wcel 2106  {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-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sn 4632
This theorem is referenced by: (None)
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