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

Theorem pwv 4627
Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
Assertion
Ref Expression
pwv 𝒫 V = V

Proof of Theorem pwv
StepHypRef Expression
1 ssv 3822 . . . 4 𝑥 ⊆ V
2 selpw 4358 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
31, 2mpbir 222 . . 3 𝑥 ∈ 𝒫 V
4 vex 3394 . . 3 𝑥 ∈ V
53, 42th 255 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2803 1 𝒫 V = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2156  Vcvv 3391  wss 3769  𝒫 cpw 4351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-in 3776  df-ss 3783  df-pw 4353
This theorem is referenced by:  univ  5109  ncanth  6833
  Copyright terms: Public domain W3C validator