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Theorem nel1nelin 41780
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 4125 . 2 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)
21con3i 157 1 𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2112  cin 3883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891
This theorem is referenced by:  nel1nelini  41782
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