Theorem elirr 5701

Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elirr	|- -. A e. A

Proof of Theorem elirr

Step	Hyp	Ref	Expression
1		id 73	. . . . 5 |- (x = A -> x = A)
2	1, 1	eleq12d 1965	. . . 4 |- (x = A -> (x e. x <-> A e. A))
3	2	notbid 673	. . 3 |- (x = A -> (-. x e. x <-> -. A e. A))
4		elirrv 5700	. . 3 |- -. x e. x
5	3, 4	vtoclg 2346	. 2 |- (A e. A -> -. A e. A)
6		pm2.01 104	. 2 |- ((A e. A -> -. A e. A) -> -. A e. A)
7	5, 6	ax-mp 7	1 |- -. A e. A

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300

This theorem is referenced by:  sucprcreg 5703  carduni 6010  alephle 6032  alephfp 6048  alephval3 6051  tpsex 8874  bnj521 12522  exnel 13869  inttarcar 15278

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049

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