Theorem pwv 4904

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 7359. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
pwv	⊢ 𝒫 V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 4005	. . . 4 ⊢ 𝑥 ⊆ V
2		velpw 4606	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
3	1, 2	mpbir 230	. . 3 ⊢ 𝑥 ∈ 𝒫 V
4		vex 3478	. . 3 ⊢ 𝑥 ∈ V
5	3, 4	2th 263	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2729	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3947  𝒫 cpw 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-pw 4603

This theorem is referenced by:  univ  5450  ncanth  7359