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Theorem elunnel2 4154
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 4152 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21biimpi 216 . . 3 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
32orcomd 871 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐶𝐴𝐵))
43orcanai 1004 1 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  wcel 2107  cun 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955
This theorem is referenced by:  fvn0fvelrn  6936  limcresiooub  45662  limcresioolb  45663  fourierdlem48  46174  fourierdlem49  46175  fourierdlem101  46227  prsal  46338  isomenndlem  46550  hsphoidmvle2  46605  hsphoidmvle  46606
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