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Theorem untelirr 23225
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 23316). 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 2313 . . . . 5  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  x ) )
2 eleq2 2314 . . . . 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 ) )
54rcla4cv 2818 . 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 1619    e. wcel 1621   A.wral 2509
This theorem is referenced by:  untsucf  23227  untangtr  23231  dfon2lem3  23309  dfon2lem7  23313  dfon2lem8  23314  dfon2lem9  23315
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729
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