MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelneq Structured version   Visualization version   GIF version

Theorem nelneq 2851
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 2815 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 239 . 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 1620  wcel 2127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1842  df-cleq 2741  df-clel 2744
This theorem is referenced by:  onfununi  7595  suc11reg  8677  cantnfp1lem3  8738  oemapvali  8742  xrge0neqmnf  12440  mreexmrid  16476  supxrnemnf  29814  onint1  32725  maxidln0  34126  rencldnfilem  37855  climlimsupcex  40473  icccncfext  40572
  Copyright terms: Public domain W3C validator