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Theorem elunsn 4683
Description: Elementhood in a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
Assertion
Ref Expression
elunsn (𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))

Proof of Theorem elunsn
StepHypRef Expression
1 elun 4153 . 2 (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 ∈ {𝐶}))
2 elsng 4640 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶))
32orbi2d 916 . 2 (𝐴𝑉 → ((𝐴𝐵𝐴 ∈ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
41, 3bitrid 283 1 (𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 848   = wceq 1540  wcel 2108  cun 3949  {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-v 3482  df-un 3956  df-sn 4627
This theorem is referenced by:  f1ounsn  7292  chnub  33002  cycpmco2lem7  33152
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