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Theorem nelneq2 2938
Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
nelneq2 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)

Proof of Theorem nelneq2
StepHypRef Expression
1 eleq2 2901 . . 3 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 250 . 2 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32con3dimp 409 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2814  df-clel 2893
This theorem is referenced by:  nelne1  3113  ssnelpss  4087  opthwiener  5396  ssfin4  9721  pwxpndom2  10076  fzneuz  12978  hauspwpwf1  22525  topdifinffinlem  34511  clsk1indlem1  40275
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