Theorem pwv 3608

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 3020	. . . 4 ⊢ 𝑥 ⊆ V
2		vex 2605	. . . . 5 ⊢ 𝑥 ∈ V
3	2	elpw 3396	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
4	1, 3	mpbir 144	. . 3 ⊢ 𝑥 ∈ 𝒫 V
5	4, 2	2th 172	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2079	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1285   ∈ wcel 1434  Vcvv 2602   ⊆ wss 2974  𝒫 cpw 3390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392

This theorem is referenced by:  univ  4233