Theorem untelirr 35433

Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 35519). 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	⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem untelirr

Step	Hyp	Ref	Expression
1		eleq1 2813	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2		eleq2 2814	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
3	1, 2	bitrd 278	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
4	3	notbid 317	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
5	4	rspccv 3603	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴))
6	5	pm2.01d 189	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051

This theorem is referenced by:  untsucf  35435  untangtr  35439  dfon2lem3  35512  dfon2lem7  35516  dfon2lem8  35517  dfon2lem9  35518