Theorem pwv 4736

Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
pwv	⊢ 𝒫 V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 3907	. . . 4 ⊢ 𝑥 ⊆ V
2		selpw 4454	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
3	1, 2	mpbir 232	. . 3 ⊢ 𝑥 ∈ 𝒫 V
4		vex 3435	. . 3 ⊢ 𝑥 ∈ V
5	3, 4	2th 265	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2790	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1520   ∈ wcel 2079  Vcvv 3432   ⊆ wss 3854  𝒫 cpw 4447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-v 3434  df-in 3861  df-ss 3869  df-pw 4449

This theorem is referenced by:  univ  5228  ncanth  6966