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Theorem elirr 7928
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 2536 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 294 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 7927 . . 3  |-  -.  x  e.  x
53, 4vtoclg 3136 . 2  |-  ( A  e.  A  ->  -.  A  e.  A )
6 pm2.01 168 . 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 1370    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-reg 7922
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-nul 3749  df-sn 3989  df-pr 3991
This theorem is referenced by:  sucprcreg  7929  sucprcregOLD  7930  alephval3  8395  rankeq1o  28376  hfninf  28391  bnj521  32083  bj-disjcsn  32797
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