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Theorem nel1nelini 44412
Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
nel1nelini.1 ¬ 𝐴𝐵
Assertion
Ref Expression
nel1nelini ¬ 𝐴 ∈ (𝐵𝐶)

Proof of Theorem nel1nelini
StepHypRef Expression
1 nel1nelini.1 . 2 ¬ 𝐴𝐵
2 nel1nelin 44410 . 2 𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))
31, 2ax-mp 5 1 ¬ 𝐴 ∈ (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  cin 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950
This theorem is referenced by: (None)
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