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

Theorem ruv 9537
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 vex 3449 . . . 4 𝑥 ∈ V
2 elirr 9532 . . . . 5 ¬ 𝑥𝑥
32nelir 3052 . . . 4 𝑥𝑥
41, 32th 263 . . 3 (𝑥 ∈ V ↔ 𝑥𝑥)
54abbi2i 2873 . 2 V = {𝑥𝑥𝑥}
65eqcomi 2745 1 {𝑥𝑥𝑥} = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  {cab 2713  wnel 3049  Vcvv 3445
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-pr 5384  ax-reg 9527
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nel 3050  df-ral 3065  df-rex 3074  df-v 3447  df-un 3915  df-sn 4587  df-pr 4589
This theorem is referenced by:  ruALT  9538
  Copyright terms: Public domain W3C validator