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Theorem pwv 4736
 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 3907 . . . 4 𝑥 ⊆ V
2 selpw 4454 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
31, 2mpbir 232 . . 3 𝑥 ∈ 𝒫 V
4 vex 3435 . . 3 𝑥 ∈ V
53, 42th 265 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2790 1 𝒫 V = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1520   ∈ wcel 2079  Vcvv 3432   ⊆ wss 3854  𝒫 cpw 4447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-v 3434  df-in 3861  df-ss 3869  df-pw 4449 This theorem is referenced by:  univ  5228  ncanth  6966
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