Theorem untelirr 32936

Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 33039). 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 2902	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2		eleq2 2903	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
3	1, 2	bitrd 281	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
4	3	notbid 320	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
5	4	rspccv 3622	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴))
6	5	pm2.01d 192	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2816  df-clel 2895  df-ral 3145

This theorem is referenced by:  untsucf  32938  untangtr  32942  dfon2lem3  33032  dfon2lem7  33036  dfon2lem8  33037  dfon2lem9  33038