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Theorem untelirr 31913
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 32023). 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 2827 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
2 eleq2 2828 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
31, 2bitrd 268 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
43notbid 307 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
54rspccv 3446 . 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 1632  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342
This theorem is referenced by:  untsucf  31915  untangtr  31919  dfon2lem3  32016  dfon2lem7  32020  dfon2lem8  32021  dfon2lem9  32022
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