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Theorem nelneq 2712
 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 2676 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 238 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32con3dimp 456 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by:  onfununi  7325  suc11reg  8399  cantnfp1lem3  8460  oemapvali  8464  xrge0neqmnf  12147  mreexmrid  16126  supxrnemnf  28924  onint1  31618  maxidln0  33014  rencldnfilem  36402  icccncfext  38773
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