Theorem pwv 3788

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 3164	. . . 4 ⊢ 𝑥 ⊆ V
2		vex 2729	. . . . 5 ⊢ 𝑥 ∈ V
3	2	elpw 3565	. . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
4	1, 3	mpbir 145	. . 3 ⊢ 𝑥 ∈ 𝒫 V
5	4, 2	2th 173	. 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
6	5	eqriv 2162	1 ⊢ 𝒫 V = V

Syntax hints:   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  𝒫 cpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  univ  4454