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Theorem elirr 7308
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
Dummy variable  x is distinct from all other variables.

Proof of Theorem elirr
StepHypRef Expression
1 id 21 . . . . 5  |-  ( x  =  A  ->  x  =  A )
21, 1eleq12d 2353 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
32notbid 287 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
4 elirrv 7307 . . 3  |-  -.  x  e.  x
53, 4vtoclg 2845 . 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 1624    e. wcel 1685
This theorem is referenced by:  sucprcreg  7309  alephval3  7733  exnel  23561  inttarcar  25301  bnj521  28033
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-reg 7302
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-v 2792  df-dif 3157  df-un 3159  df-nul 3458  df-sn 3648  df-pr 3649
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