Theorem pwv 3892

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 3249	. . . 4 ⊢ 𝑥 ⊆ V
2		vex 2805	. . . . 5 ⊢ 𝑥 ∈ V
3	2	elpw 3658	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
4	1, 3	mpbir 146	. . 3 ⊢ 𝑥 ∈ 𝒫 V
5	4, 2	2th 174	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2228	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  𝒫 cpw 3652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  univ  4573