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Theorem nelneq 2902
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 2866 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 241 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32con3dimp 398 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
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 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792  df-clel 2795
This theorem is referenced by:  onfununi  7677  suc11reg  8766  cantnfp1lem3  8827  oemapvali  8831  xrge0neqmnfOLD  12527  mreexmrid  16618  supxrnemnf  30052  onint1  32956  maxidln0  34331  rencldnfilem  38170  climlimsupcex  40745  icccncfext  40844
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