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

Theorem untelirr 32186
 Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 32289). 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 2847 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
2 eleq2 2848 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
31, 2bitrd 271 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
43notbid 310 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
54rspccv 3508 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝐴𝐴 → ¬ 𝐴𝐴))
65pm2.01d 182 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1601   ∈ wcel 2107  ∀wral 3090 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-v 3400 This theorem is referenced by:  untsucf  32188  untangtr  32192  dfon2lem3  32282  dfon2lem7  32286  dfon2lem8  32287  dfon2lem9  32288
 Copyright terms: Public domain W3C validator