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Theorem elirr 7245
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
StepHypRef Expression
1 id 21 . . . . 5  |-  ( x  =  A  ->  x  =  A )
21, 1eleq12d 2324 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 287 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 7244 . . 3  |-  -.  x  e.  x
53, 4vtoclg 2794 . 2  |-  ( A  e.  A  ->  -.  A  e.  A )
6 pm2.01 162 . 2  |-  ( ( A  e.  A  ->  -.  A  e.  A
)  ->  -.  A  e.  A )
75, 6ax-mp 10 1  |-  -.  A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621
This theorem is referenced by:  sucprcreg  7246  alephval3  7670  exnel  23493  inttarcar  25233  bnj521  27777
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-reg 7239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-v 2742  df-dif 3097  df-un 3099  df-nul 3398  df-sn 3587  df-pr 3588
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