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Theorem ruv 8368
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 3174 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 1925 . . . 4 𝑥 = 𝑥
3 elirrv 8365 . . . . 5 ¬ 𝑥𝑥
43nelir 2885 . . . 4 𝑥𝑥
52, 42th 252 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2725 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2632 1 {𝑥𝑥𝑥} = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  {cab 2595  wnel 2780  Vcvv 3172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-reg 8358
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-nel 2782  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-nul 3874  df-sn 4125  df-pr 4127
This theorem is referenced by:  ruALT  8369
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