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Theorem elirr 7328
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 2364 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 285 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 7327 . . 3  |-  -.  x  e.  x
53, 4vtoclg 2856 . 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 1632    e. wcel 1696
This theorem is referenced by:  sucprcreg  7329  alephval3  7753  exnel  24230  inttarcar  26004  bnj521  29081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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