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Theorem untelirr 25159
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 25421). 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 2498 . . . . 5  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  x ) )
2 eleq2 2499 . . . . 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 3051 . 2  |-  ( A. x  e.  A  -.  x  e.  x  ->  ( A  e.  A  ->  -.  A  e.  A
) )
65pm2.01d 164 1  |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707
This theorem is referenced by:  untsucf  25161  untangtr  25165  dfon2lem3  25414  dfon2lem7  25418  dfon2lem8  25419  dfon2lem9  25420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960
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