Theorem pwv 4574

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 7376. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
pwv	⊢ 𝒫 V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 3932	. . . 4 ⊢ 𝑥 ⊆ V
2		velpw 4561	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
3	1, 2	mpbir 234	. . 3 ⊢ 𝑥 ∈ 𝒫 V
4		vex 3454	. . 3 ⊢ 𝑥 ∈ V
5	3, 4	2th 267	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2757	1 ⊢ 𝒫 V = V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-pw 4558

This theorem is used by:  univ  5418  ncanth  7376