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Theorem ruv 8545
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 3233 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 1985 . . . 4 𝑥 = 𝑥
3 elirrv 8542 . . . . 5 ¬ 𝑥𝑥
43nelir 2929 . . . 4 𝑥𝑥
52, 42th 254 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2768 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2674 1 {𝑥𝑥𝑥} = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  {cab 2637  wnel 2926  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  ruALT  8546
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