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Theorem elunsn 30258
 Description: Elementhood to 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 4101 . 2 (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 ∈ {𝐶}))
2 elsng 4554 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶))
32orbi2d 913 . 2 (𝐴𝑉 → ((𝐴𝐵𝐴 ∈ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
41, 3syl5bb 286 1 (𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2115   ∪ cun 3908  {csn 4540 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-un 3915  df-sn 4541 This theorem is referenced by:  cycpmco2lem7  30781
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