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Theorem neldif 4143
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 3972 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
21simplbi2 500 . . 3 (𝐴𝐵 → (¬ 𝐴𝐶𝐴 ∈ (𝐵𝐶)))
32con1d 145 . 2 (𝐴𝐵 → (¬ 𝐴 ∈ (𝐵𝐶) → 𝐴𝐶))
43imp 406 1 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵𝐶)) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  cdif 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965
This theorem is referenced by:  peano5  7915  boxcutc  8979  etransc  46238
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