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Theorem untelirr 32959
Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 33062). 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 2903 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
2 eleq2 2904 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
31, 2bitrd 282 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
43notbid 321 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
54rspccv 3606 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝐴𝐴 → ¬ 𝐴𝐴))
65pm2.01d 193 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  wral 3133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-ral 3138
This theorem is referenced by:  untsucf  32961  untangtr  32965  dfon2lem3  33055  dfon2lem7  33059  dfon2lem8  33060  dfon2lem9  33061
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