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Theorem elunnel1 3716
 Description: A member of a union that is not member of the first class, is member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elunnel1 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐵) → 𝐴𝐶)

Proof of Theorem elunnel1
StepHypRef Expression
1 elun 3715 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21biimpi 205 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
32orcanai 950 1 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐵) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∈ wcel 1977   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545 This theorem is referenced by:  fprodsplitsn  14559  founiiun0  38372  fsumsplitsn  38637  dvnprodlem1  38836  fourierdlem70  39069  fourierdlem71  39070  fourierdlem80  39079  sge0splitmpt  39304  sge0iunmptlemfi  39306  nnfoctbdjlem  39348  hoidmvlelem2  39486  hoidmvlelem3  39487  pimrecltpos  39596
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