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Theorem untelirr 23459
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 23550). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
untelirr  |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
Distinct variable group:    x, A

Proof of Theorem untelirr
StepHypRef Expression
1 eleq1 2345 . . . . 5  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  x ) )
2 eleq2 2346 . . . . 5  |-  ( x  =  A  ->  ( A  e.  x  <->  A  e.  A ) )
31, 2bitrd 246 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
43notbid 287 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
54rspccv 2883 . 2  |-  ( A. x  e.  A  -.  x  e.  x  ->  ( A  e.  A  ->  -.  A  e.  A
) )
65pm2.01d 163 1  |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1624    e. wcel 1685   A.wral 2545
This theorem is referenced by:  untsucf  23461  untangtr  23465  dfon2lem3  23543  dfon2lem7  23547  dfon2lem8  23548  dfon2lem9  23549
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-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
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-ral 2550  df-v 2792
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