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Theorem nelneq 2941
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 2904 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 250 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32con3dimp 409 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2818  df-clel 2897
This theorem is referenced by:  nelne2  3119  onfununi  7972  suc11reg  9074  cantnfp1lem3  9135  oemapvali  9139  mreexmrid  16906  supxrnemnf  30406  onint1  33681  bj-fvmptunsn2  34419  maxidln0  35191  rencldnfilem  39278  climlimsupcex  41911  icccncfext  42031
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