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Theorem untelirr 23412
 Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 23503). 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 2316 . . . . 5
2 eleq2 2317 . . . . 5
31, 2bitrd 246 . . . 4
43notbid 287 . . 3
54rcla4cv 2849 . 2
65pm2.01d 163 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wceq 1619   wcel 1621  wral 2516 This theorem is referenced by:  untsucf  23414  untangtr  23418  dfon2lem3  23496  dfon2lem7  23500  dfon2lem8  23501  dfon2lem9  23502 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 1927  ax-ext 2237 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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2521  df-v 2759
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