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

Theorem pwv 4465
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 3658 . . . 4 𝑥 ⊆ V
2 selpw 4198 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
31, 2mpbir 221 . . 3 𝑥 ∈ 𝒫 V
4 vex 3234 . . 3 𝑥 ∈ V
53, 42th 254 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2648 1 𝒫 V = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193
This theorem is referenced by:  univ  4949  ncanth  6649
  Copyright terms: Public domain W3C validator