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Theorem ruv 8390
 Description: The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
Assertion
Ref Expression
ruv {𝑥𝑥𝑥} = V

Proof of Theorem ruv
StepHypRef Expression
1 df-v 3175 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 1926 . . . 4 𝑥 = 𝑥
3 elirrv 8387 . . . . 5 ¬ 𝑥𝑥
43nelir 2886 . . . 4 𝑥𝑥
52, 42th 253 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2726 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2633 1 {𝑥𝑥𝑥} = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  {cab 2596   ∉ wnel 2781  Vcvv 3173 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-nel 2783  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:  ruALT  8391
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