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Theorem elirr 9635
Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elirr ¬ 𝐴𝐴

Proof of Theorem elirr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
21, 1eleq12d 2833 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
32notbid 318 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4 elirrv 9634 . . 3 ¬ 𝑥𝑥
53, 4vtoclg 3554 . 2 (𝐴𝐴 → ¬ 𝐴𝐴)
6 pm2.01 188 . 2 ((𝐴𝐴 → ¬ 𝐴𝐴) → ¬ 𝐴𝐴)
75, 6ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by:  elneq  9636  sucprcreg  9639  ruv  9640  disjcsn  9642  alephval3  10148  prv1n  35416  rankeq1o  36153  hfninf  36168  bj-iomnnom  37242  omabs2  43322
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