Theorem elirr 8505

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 2695	. . . 4 |- ( x = A -> ( x e. x <-> A e. A ) )
3	2	notbid 308	. . 3 |- ( x = A -> ( -. x e. x <-> -. A e. A ) )
4		elirrv 8504	. . 3 |- -. x e. x
5	3, 4	vtoclg 3266	. 2 |- ( A e. A -> -. A e. A )
6		pm2.01 180	. 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 1483   e. wcel 1990

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  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

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-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  sucprcreg  8506  alephval3  8933  n0lplig  27335  bnj521  30805  rankeq1o  32278  hfninf  32293  bj-disjcsn  32936