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Theorem elirr 9243
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 2834 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
32notbid 321 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4 elirrv 9242 . . 3 ¬ 𝑥𝑥
53, 4vtoclg 3496 . 2 (𝐴𝐴 → ¬ 𝐴𝐴)
6 pm2.01 192 . 2 ((𝐴𝐴 → ¬ 𝐴𝐴) → ¬ 𝐴𝐴)
75, 6ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pr 5339  ax-reg 9238
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-v 3425  df-dif 3886  df-un 3888  df-nul 4255  df-sn 4559  df-pr 4561
This theorem is referenced by:  elneq  9244  sucprcreg  9247  ruv  9248  alephval3  9754  bnj521  32460  prv1n  33137  rankeq1o  34244  hfninf  34259  bj-disjcsn  34912  bj-iomnnom  35201
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