Theorem elirr 7928

Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elirr	|- -. A e. A

Proof of Theorem elirr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 |- ( x = A -> x = A )
2	1, 1	eleq12d 2536	. . . 4 |- ( x = A -> ( x e. x <-> A e. A ) )
3	2	notbid 294	. . 3 |- ( x = A -> ( -. x e. x <-> -. A e. A ) )
4		elirrv 7927	. . 3 |- -. x e. x
5	3, 4	vtoclg 3136	. 2 |- ( A e. A -> -. A e. A )
6		pm2.01 168	. 2 |- ( ( A e. A -> -. A e. A ) -> -. A e. A )
7	5, 6	ax-mp 5	1 |- -. A e. A

Syntax hints:   -. wn 3   -> wi 4   = wceq 1370   e. wcel 1758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-reg 7922

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-nul 3749  df-sn 3989  df-pr 3991

This theorem is referenced by:  sucprcreg  7929  sucprcregOLD  7930  alephval3  8395  rankeq1o  28376  hfninf  28391  bnj521  32083  bj-disjcsn  32797