Theorem elirr 8042

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 2539	. . . 4 |- ( x = A -> ( x e. x <-> A e. A ) )
3	2	notbid 294	. . 3 |- ( x = A -> ( -. x e. x <-> -. A e. A ) )
4		elirrv 8041	. . 3 |- -. x e. x
5	3, 4	vtoclg 3167	. 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 1395   e. wcel 1819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-nul 3794  df-sn 4033  df-pr 4035

This theorem is referenced by:  sucprcreg  8043  alephval3  8508  rankeq1o  30033  hfninf  30048  bnj521  33935  bj-disjcsn  34648