Theorem untelirr 31585

Description: We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 31697). 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	|- ( A. x e. A -. x e. x -> -. A e. A )

Distinct variable group:   x, A

Proof of Theorem untelirr

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( x = A -> ( x e. x <-> A e. x ) )
2		eleq2 2690	. . . . 5 |- ( x = A -> ( A e. x <-> A e. A ) )
3	1, 2	bitrd 268	. . . 4 |- ( x = A -> ( x e. x <-> A e. A ) )
4	3	notbid 308	. . 3 |- ( x = A -> ( -. x e. x <-> -. A e. A ) )
5	4	rspccv 3306	. 2 |- ( A. x e. A -. x e. x -> ( A e. A -> -. A e. A ) )
6	5	pm2.01d 181	1 |- ( A. x e. A -. x e. x -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  untsucf  31587  untangtr  31591  dfon2lem3  31690  dfon2lem7  31694  dfon2lem8  31695  dfon2lem9  31696