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Theorem elirr 8042
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 2539 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 294 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 8041 . . 3  |-  -.  x  e.  x
53, 4vtoclg 3167 . 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 1395    e. wcel 1819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-nul 3794  df-sn 4033  df-pr 4035
This theorem is referenced by:  sucprcreg  8043  alephval3  8508  rankeq1o  30033  hfninf  30048  bnj521  33935  bj-disjcsn  34648
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