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Theorem nelneq2 2904
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 2868 . . 3 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 241 . 2 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32con3dimp 398 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2793  df-clel 2796
This theorem is referenced by:  ssnelpss  3916  opthwiener  5171  ssfin4  9421  pwxpndom2  9776  fzneuz  12674  hauspwpwf1  22118  topdifinffinlem  33692  clsk1indlem1  39120
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