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Theorem elunnel1 4163
Description: A member of a union that is not a member of the first class, is a 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 4162 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21biimpi 216 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
32orcanai 1004 1 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐵) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  wcel 2105  cun 3960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967
This theorem is referenced by:  fsumsplitsn  15776  fprodsplitsn  16021  founiiun0  45132  infxrpnf  45395  cnrefiisplem  45784  dvnprodlem1  45901  fourierdlem70  46131  fourierdlem71  46132  fourierdlem80  46141  sge0splitmpt  46366  sge0iunmptlemfi  46368  nnfoctbdjlem  46410  hoidmvlelem2  46551  hoidmvlelem3  46552  pimrecltpos  46663
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