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Theorem pwv 4585
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 3766 . . . 4 𝑥 ⊆ V
2 selpw 4309 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
31, 2mpbir 221 . . 3 𝑥 ∈ 𝒫 V
4 vex 3343 . . 3 𝑥 ∈ V
53, 42th 254 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2757 1 𝒫 V = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304
This theorem is referenced by:  univ  5068  ncanth  6773
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