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

Theorem elirr 8118
Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elirr  |-  -.  A  e.  A

Proof of Theorem elirr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
21, 1eleq12d 2525 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 296 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 8117 . . 3  |-  -.  x  e.  x
53, 4vtoclg 3109 . 2  |-  ( A  e.  A  ->  -.  A  e.  A )
6 pm2.01 172 . 2  |-  ( ( A  e.  A  ->  -.  A  e.  A
)  ->  -.  A  e.  A )
75, 6ax-mp 5 1  |-  -.  A  e.  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1446    e. wcel 1889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-reg 8112
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-v 3049  df-dif 3409  df-un 3411  df-nul 3734  df-sn 3971  df-pr 3973
This theorem is referenced by:  sucprcreg  8119  alephval3  8546  bnj521  29557  rankeq1o  30950  hfninf  30965  bj-disjcsn  31554
  Copyright terms: Public domain W3C validator