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

Theorem elneq 9050
Description: A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022.)
Assertion
Ref Expression
elneq (𝐴𝐵𝐴𝐵)

Proof of Theorem elneq
StepHypRef Expression
1 elirr 9049 . 2 ¬ 𝐵𝐵
2 nelelne 3114 . 2 𝐵𝐵 → (𝐴𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2105  wne 3013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  nelaneq  9051  preleqg  9066  dfac2b  9544
  Copyright terms: Public domain W3C validator