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Theorem neldif 3879
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 3726 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
21simplbi2 656 . . 3 (𝐴𝐵 → (¬ 𝐴𝐶𝐴 ∈ (𝐵𝐶)))
32con1d 139 . 2 (𝐴𝐵 → (¬ 𝐴 ∈ (𝐵𝐶) → 𝐴𝐶))
43imp 444 1 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵𝐶)) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2140  cdif 3713
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
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 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-dif 3719
This theorem is referenced by:  peano5  7256  boxcutc  8120  etransc  41022
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