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Theorem nelne1 2398
 Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nelne1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Proof of Theorem nelne1
StepHypRef Expression
1 eleq2 2203 . . . 4 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 158 . . 3 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32necon3bd 2351 . 2 (𝐴𝐵 → (¬ 𝐴𝐶𝐵𝐶))
43imp 123 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ≠ wne 2308 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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-ne 2309 This theorem is referenced by:  elnelne1  2412  difsnb  3663
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