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Theorem pwv 4366
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 3588 . . . 4 𝑥 ⊆ V
2 selpw 4115 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
31, 2mpbir 220 . . 3 𝑥 ∈ 𝒫 V
4 vex 3176 . . 3 𝑥 ∈ V
53, 42th 253 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2607 1 𝒫 V = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540  𝒫 cpw 4108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110
This theorem is referenced by:  univ  4840  ncanth  6487
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