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

Theorem elirr 9592
Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elirr ¬ 𝐴𝐴

Proof of Theorem elirr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
21, 1eleq12d 2828 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
32notbid 318 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4 elirrv 9591 . . 3 ¬ 𝑥𝑥
53, 4vtoclg 3557 . 2 (𝐴𝐴 → ¬ 𝐴𝐴)
6 pm2.01 188 . 2 ((𝐴𝐴 → ¬ 𝐴𝐴) → ¬ 𝐴𝐴)
75, 6ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-pr 5428  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632
This theorem is referenced by:  elneq  9593  sucprcreg  9596  ruv  9597  disjcsn  9599  alephval3  10105  prv1n  34422  rankeq1o  35143  hfninf  35158  bj-iomnnom  36140  omabs2  42082
  Copyright terms: Public domain W3C validator