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Theorem untelirr 30343
 Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 30445). 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 2495 . . . . 5
2 eleq2 2496 . . . . 5
31, 2bitrd 256 . . . 4
43notbid 295 . . 3
54rspccv 3179 . 2
65pm2.01d 172 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1437   wcel 1872  wral 2771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-v 3082 This theorem is referenced by:  untsucf  30345  untangtr  30349  dfon2lem3  30438  dfon2lem7  30442  dfon2lem8  30443  dfon2lem9  30444
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