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Theorem pwv 4903
Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235.

The collection of all classes is of course larger than V, which is the collection of all sets. But 𝒫 V, being a class, cannot contain proper classes, so 𝒫 V is actually no larger than V. This fact is exploited in ncanth 7387. (Contributed by NM, 14-Sep-2003.)

Assertion
Ref Expression
pwv 𝒫 V = V

Proof of Theorem pwv
StepHypRef Expression
1 ssv 4007 . . . 4 𝑥 ⊆ V
2 velpw 4604 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
31, 2mpbir 231 . . 3 𝑥 ∈ 𝒫 V
4 vex 3483 . . 3 𝑥 ∈ V
53, 42th 264 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2733 1 𝒫 V = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950  𝒫 cpw 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-pw 4601
This theorem is referenced by:  univ  5455  ncanth  7387
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