Theorem pwv 3887

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	⊢ ℘V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 3292	. . . 4 ⊢ x ⊆ V
2		vex 2863	. . . . 5 ⊢ x ∈ V
3	2	elpw 3729	. . . 4 ⊢ (x ∈ ℘V ↔ x ⊆ V)
4	1, 3	mpbir 200	. . 3 ⊢ x ∈ ℘V
5	4, 2	2th 230	. 2 ⊢ (x ∈ ℘V ↔ x ∈ V)
6	5	eqriv 2350	1 ⊢ ℘V = V

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ℘cpw 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725

This theorem is referenced by:  1cvsfin  4543  ncpw1c  6155  ce2nc1  6194  nchoicelem19  6308  vncan  6338