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

Theorem ruv 9642
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 3484 . . . 4 𝑥 ∈ V
2 elirr 9637 . . . . 5 ¬ 𝑥𝑥
32nelir 3049 . . . 4 𝑥𝑥
41, 32th 264 . . 3 (𝑥 ∈ V ↔ 𝑥𝑥)
54eqabi 2877 . 2 V = {𝑥𝑥𝑥}
65eqcomi 2746 1 {𝑥𝑥𝑥} = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  wnel 3046  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  ruALT  9643
  Copyright terms: Public domain W3C validator