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Theorem elirr 8751
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 ¬ 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 2890 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
32notbid 309 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4 elirrv 8750 . . 3 ¬ 𝑥𝑥
53, 4vtoclg 3470 . 2 (𝐴𝐴 → ¬ 𝐴𝐴)
6 pm2.01 180 . 2 ((𝐴𝐴 → ¬ 𝐴𝐴) → ¬ 𝐴𝐴)
75, 6ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1637  wcel 2157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109  ax-reg 8746
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-dif 3783  df-un 3785  df-nul 4128  df-sn 4382  df-pr 4384
This theorem is referenced by:  elneq  8752  sucprcreg  8755  alephval3  9226  n0lplig  27689  bnj521  31151  rankeq1o  32621  hfninf  32636  bj-disjcsn  33266
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