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Theorem nelne2 2296
 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 2100 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 148 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 2248 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 115 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ≠ wne 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-ne 2206 This theorem is referenced by: (None)
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