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Theorem untelirr 25110
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 25362). 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 2464 . . . . 5  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  x ) )
2 eleq2 2465 . . . . 5  |-  ( x  =  A  ->  ( A  e.  x  <->  A  e.  A ) )
31, 2bitrd 245 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
43notbid 286 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
54rspccv 3009 . 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 3    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2666
This theorem is referenced by:  untsucf  25112  untangtr  25116  dfon2lem3  25355  dfon2lem7  25359  dfon2lem8  25360  dfon2lem9  25361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918
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