Theorem elirr 8022

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 2523	. . . 4 |- ( x = A -> ( x e. x <-> A e. A ) )
3	2	notbid 294	. . 3 |- ( x = A -> ( -. x e. x <-> -. A e. A ) )
4		elirrv 8021	. . 3 |- -. x e. x
5	3, 4	vtoclg 3151	. 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 1381   e. wcel 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-reg 8016

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-v 3095  df-dif 3461  df-un 3463  df-nul 3768  df-sn 4011  df-pr 4013

This theorem is referenced by:  sucprcreg  8023  sucprcregOLD  8024  alephval3  8489  rankeq1o  29796  hfninf  29811  bnj521  33500  bj-disjcsn  34217