Theorem elirr 8118

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 2525	. . . 4 |- ( x = A -> ( x e. x <-> A e. A ) )
3	2	notbid 296	. . 3 |- ( x = A -> ( -. x e. x <-> -. A e. A ) )
4		elirrv 8117	. . 3 |- -. x e. x
5	3, 4	vtoclg 3109	. 2 |- ( A e. A -> -. A e. A )
6		pm2.01 172	. 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 1446   e. wcel 1889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-reg 8112

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-v 3049  df-dif 3409  df-un 3411  df-nul 3734  df-sn 3971  df-pr 3973

This theorem is referenced by:  sucprcreg  8119  alephval3  8546  bnj521  29557  rankeq1o  30950  hfninf  30965  bj-disjcsn  31554