ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nelneq2 GIF version

Theorem nelneq2 2155
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 2117 . . 3 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 152 . 2 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32con3dimp 574 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  ssnelpss  3316  dtruarb  3969  fzneuz  9064
  Copyright terms: Public domain W3C validator