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Theorem elunnel2 39720
Description: A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elunnel2 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)

Proof of Theorem elunnel2
StepHypRef Expression
1 elun 3904 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21biimpi 206 . . 3 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
32orcomd 860 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐶𝐴𝐵))
43orcanai 987 1 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 836  wcel 2145  cun 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728
This theorem is referenced by:  limcresiooub  40389  limcresioolb  40390  fourierdlem48  40885  fourierdlem49  40886  fourierdlem101  40938  prsal  41052  isomenndlem  41261  hsphoidmvle2  41316  hsphoidmvle  41317
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