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Theorem elunsn 32520
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 4163 . 2 (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 ∈ {𝐶}))
2 elsng 4645 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶))
32orbi2d 914 . 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 846   = wceq 1535  wcel 2104  cun 3961  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479  df-un 3968  df-sn 4632
This theorem is referenced by:  chnub  32962  cycpmco2lem7  33103
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