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Theorem elirr 8388
 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 2682 . . . 4 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
32notbid 307 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4 elirrv 8387 . . 3 ¬ 𝑥𝑥
53, 4vtoclg 3239 . 2 (𝐴𝐴 → ¬ 𝐴𝐴)
6 pm2.01 179 . 2 ((𝐴𝐴 → ¬ 𝐴𝐴) → ¬ 𝐴𝐴)
75, 6ax-mp 5 1 ¬ 𝐴𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  sucprcreg  8389  alephval3  8816  bnj521  30059  rankeq1o  31448  hfninf  31463  bj-disjcsn  32129
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