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	⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem untelirr

Step	Hyp	Ref	Expression
1		eleq1 2847	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2		eleq2 2848	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
3	1, 2	bitrd 271	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
4	3	notbid 310	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
5	4	rspccv 3508	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴))
6	5	pm2.01d 182	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)

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