Theorem pwv 4864

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 7355. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
pwv	⊢ 𝒫 V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 3963	. . . 4 ⊢ 𝑥 ⊆ V
2		velpw 4563	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
3	1, 2	mpbir 234	. . 3 ⊢ 𝑥 ∈ 𝒫 V
4		vex 3461	. . 3 ⊢ 𝑥 ∈ V
5	3, 4	2th 267	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2762	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907  𝒫 cpw 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3924  df-pw 4560

This theorem is referenced by:  univ  5422  ncanth  7355