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Theorem untelirr 30682
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 30783). 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 2580 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
2 eleq2 2581 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
31, 2bitrd 266 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
43notbid 306 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
54rspccv 3183 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝐴𝐴 → ¬ 𝐴𝐴))
65pm2.01d 179 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wcel 1938  wral 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-v 3079
This theorem is referenced by:  untsucf  30684  untangtr  30688  dfon2lem3  30776  dfon2lem7  30780  dfon2lem8  30781  dfon2lem9  30782
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