Theorem pwv 4871

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 7345. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
pwv	⊢ 𝒫 V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 3974	. . . 4 ⊢ 𝑥 ⊆ V
2		velpw 4571	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
3	1, 2	mpbir 231	. . 3 ⊢ 𝑥 ∈ 𝒫 V
4		vex 3454	. . 3 ⊢ 𝑥 ∈ V
5	3, 4	2th 264	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2727	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  𝒫 cpw 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-pw 4568

This theorem is referenced by:  univ  5414  ncanth  7345