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Theorem neldif 3990
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 3833 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
21simplbi2 493 . . 3 (𝐴𝐵 → (¬ 𝐴𝐶𝐴 ∈ (𝐵𝐶)))
32con1d 142 . 2 (𝐴𝐵 → (¬ 𝐴 ∈ (𝐵𝐶) → 𝐴𝐶))
43imp 398 1 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵𝐶)) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wcel 2050  cdif 3820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-dif 3826
This theorem is referenced by:  peano5  7414  boxcutc  8296  etransc  41999
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