MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruv Structured version   Visualization version   GIF version

Theorem ruv 9054
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 3446 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 2019 . . . 4 𝑥 = 𝑥
3 elirrv 9048 . . . . 5 ¬ 𝑥𝑥
43nelir 3097 . . . 4 𝑥𝑥
52, 42th 267 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2866 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2825 1 {𝑥𝑥𝑥} = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  {cab 2779  wnel 3094  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-reg 9044
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nel 3095  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-nul 4247  df-sn 4529  df-pr 4531
This theorem is referenced by:  ruALT  9055
  Copyright terms: Public domain W3C validator