Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elunnel2 Structured version   Visualization version   GIF version

Theorem elunnel2 39018
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 3745 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21biimpi 206 . . 3 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
32orcomd 403 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐶𝐴𝐵))
43orcanai 951 1 ((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  wcel 1988  cun 3565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572
This theorem is referenced by:  limcresiooub  39674  limcresioolb  39675  fourierdlem48  40134  fourierdlem49  40135  fourierdlem101  40187  prsal  40301  isomenndlem  40507  hsphoidmvle2  40562  hsphoidmvle  40563
  Copyright terms: Public domain W3C validator