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Theorem bj-elsn12g 34779
 Description: Join of elsng 4536 and elsn2g 4560. (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 4536 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 elsn2g 4560 . 2 (𝐵𝑊 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
31, 2jaoi 854 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {csn 4522 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-sn 4523 This theorem is referenced by: (None)
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