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Theorem nel1nelin 39651
Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Assertion
Ref Expression
nel1nelin 𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))

Proof of Theorem nel1nelin
StepHypRef Expression
1 elinel1 3832 . 2 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)
21con3i 150 1 𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2030  cin 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614
This theorem is referenced by:  nel1nelini  39654
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