Theorem elirr 5934

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 15	. . . . 5 |- (x = A -> x = A)
2	1, 1	eleq12d 2212	. . . 4 |- (x = A -> (x e. x <-> A e. A))
3	2	notbid 746	. . 3 |- (x = A -> (-. x e. x <-> -. A e. A))
4		elirrv 5933	. . 3 |- -. x e. x
5	3, 4	vtoclg 2588	. 2 |- (A e. A -> -. A e. A)
6		pm2.01 210	. 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 1586   e. wcel 1588

This theorem is referenced by:  sucprcreg 5935  alephle 6213  alephfp 6231  alephval3 6233  carduniOLD 6424  tpsexOLD 9711  bnj521 13295  exnel 14510  inttarcar 16088

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-14 1600  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-sep 3606  ax-nul 3613  ax-pow 3649  ax-reg 5928

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-ral 2359  df-rex 2360  df-v 2540  df-dif 2830  df-in 2834  df-ss 2836  df-nul 3083  df-pw 3229  df-sn 3242

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