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Theorem nelneq 2722
 Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
Assertion
Ref Expression
nelneq ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)

Proof of Theorem nelneq
StepHypRef Expression
1 eleq1 2686 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 239 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32con3dimp 457 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-cleq 2614  df-clel 2617 This theorem is referenced by:  onfununi  7390  suc11reg  8467  cantnfp1lem3  8528  oemapvali  8532  xrge0neqmnf  12225  mreexmrid  16231  supxrnemnf  29396  onint1  32117  maxidln0  33503  rencldnfilem  36891  icccncfext  39426
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