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Theorem nelneq2 2864
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 2827 . . 3 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 248 . 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 1539  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-clel 2816
This theorem is referenced by:  nelne1  3041  ssnelpss  4046  opthwiener  5428  ssfin4  10066  pwxpndom2  10421  fzneuz  13337  hauspwpwf1  23138  topdifinffinlem  35518  clsk1indlem1  41655
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