Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  untelirr Structured version   Unicode version

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  |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
Distinct variable group:    x, A

Proof of Theorem untelirr
StepHypRef Expression
1 eleq1 2495 . . . . 5  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  x ) )
2 eleq2 2496 . . . . 5  |-  ( x  =  A  ->  ( A  e.  x  <->  A  e.  A ) )
31, 2bitrd 256 . . . 4  |-  ( x  =  A  ->  (
x  e.  x  <->  A  e.  A ) )
43notbid 295 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  x  <->  -.  A  e.  A ) )
54rspccv 3179 . 2  |-  ( A. x  e.  A  -.  x  e.  x  ->  ( A  e.  A  ->  -.  A  e.  A
) )
65pm2.01d 172 1  |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1872   A.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
  Copyright terms: Public domain W3C validator