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Theorem nelne2 2400
 Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
Assertion
Ref Expression
nelne2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Proof of Theorem nelne2
StepHypRef Expression
1 eleq1 2203 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 158 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 2352 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 123 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   ≠ wne 2309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136  df-ne 2310 This theorem is referenced by:  nelelne  2401  elnelne2  2414  zgt1rpn0n1  9532
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