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Theorem nel2nelin 39837
Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Assertion
Ref Expression
nel2nelin 𝐴𝐶 → ¬ 𝐴 ∈ (𝐵𝐶))

Proof of Theorem nel2nelin
StepHypRef Expression
1 elinel2 3943 . 2 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)
21con3i 150 1 𝐴𝐶 → ¬ 𝐴 ∈ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2139  cin 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722
This theorem is referenced by:  nel2nelini  39840
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