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Theorem nel1nelini 39839
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 39836 . 2 𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))
31, 2ax-mp 5 1 ¬ 𝐴 ∈ (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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: (None)
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