MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelneq2 Structured version   Visualization version   GIF version

Theorem nelneq2 2864
Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
nelneq2 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)

Proof of Theorem nelneq2
StepHypRef Expression
1 eleq2 2828 . . 3 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 239 . 2 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32con3dimp 456 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756
This theorem is referenced by:  ssnelpss  3860  opthwiener  5124  ssfin4  9344  pwxpndom2  9699  fzneuz  12634  hauspwpwf1  22012  topdifinffinlem  33524  clsk1indlem1  38863
  Copyright terms: Public domain W3C validator