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

Theorem untelirr 31884
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 31994). 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 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem untelirr
StepHypRef Expression
1 eleq1 2819 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
2 eleq2 2820 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
31, 2bitrd 268 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
43notbid 307 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
54rspccv 3438 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝐴𝐴 → ¬ 𝐴𝐴))
65pm2.01d 181 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1624  wcel 2131  wral 3042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-v 3334
This theorem is referenced by:  untsucf  31886  untangtr  31890  dfon2lem3  31987  dfon2lem7  31991  dfon2lem8  31992  dfon2lem9  31993
  Copyright terms: Public domain W3C validator