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Theorem elirr 7312
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 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
21, 1eleq12d 2351 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 285 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 7311 . . 3  |-  -.  x  e.  x
53, 4vtoclg 2843 . 2  |-  ( A  e.  A  ->  -.  A  e.  A )
6 pm2.01 160 . 2  |-  ( ( A  e.  A  ->  -.  A  e.  A
)  ->  -.  A  e.  A )
75, 6ax-mp 8 1  |-  -.  A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684
This theorem is referenced by:  sucprcreg  7313  alephval3  7737  exnel  24159  inttarcar  25901  bnj521  28765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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