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

Theorem nonconne 2835
Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
Assertion
Ref Expression
nonconne ¬ (𝐴 = 𝐵𝐴𝐵)

Proof of Theorem nonconne
StepHypRef Expression
1 fal 1530 . 2 ¬ ⊥
2 eqneqall 2834 . . 3 (𝐴 = 𝐵 → (𝐴𝐵 → ⊥))
32imp 444 . 2 ((𝐴 = 𝐵𝐴𝐵) → ⊥)
41, 3mto 188 1 ¬ (𝐴 = 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wfal 1528  wne 2823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-fal 1529  df-ne 2824
This theorem is referenced by:  osumcllem11N  35570  pexmidlem8N  35581  dochexmidlem8  37073
  Copyright terms: Public domain W3C validator