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Theorem untelirr 24054
 Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 24148). 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
Distinct variable group:   ,

Proof of Theorem untelirr
StepHypRef Expression
1 eleq1 2343 . . . . 5
2 eleq2 2344 . . . . 5
31, 2bitrd 244 . . . 4
43notbid 285 . . 3
54rspccv 2881 . 2
65pm2.01d 161 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1623   wcel 1684  wral 2543 This theorem is referenced by:  untsucf  24056  untangtr  24060  dfon2lem3  24141  dfon2lem7  24145  dfon2lem8  24146  dfon2lem9  24147 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
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