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Theorem neldif 4092
 Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
Assertion
Ref Expression
neldif ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵𝐶)) → 𝐴𝐶)

Proof of Theorem neldif
StepHypRef Expression
1 eldif 3929 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
21simplbi2 504 . . 3 (𝐴𝐵 → (¬ 𝐴𝐶𝐴 ∈ (𝐵𝐶)))
32con1d 147 . 2 (𝐴𝐵 → (¬ 𝐴 ∈ (𝐵𝐶) → 𝐴𝐶))
43imp 410 1 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵𝐶)) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2115   ∖ cdif 3916 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-dif 3922 This theorem is referenced by:  peano5  7599  boxcutc  8501  etransc  42851
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