Theorem pwv 3891

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 3248	. . . 4 |- x C_ _V
2		vex 2804	. . . . 5 |- x e. _V
3	2	elpw 3657	. . . 4 |- ( x e. ~P _V <-> x C_ _V )
4	1, 3	mpbir 146	. . 3 |- x e. ~P _V
5	4, 2	2th 174	. 2 |- ( x e. ~P _V <-> x e. _V )
6	5	eqriv 2227	1 |- ~P _V = _V

Syntax hints:   = wceq 1397   e. wcel 2201   _Vcvv 2801   C_ wss 3199   ~Pcpw 3651

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 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-pw 3653

This theorem is referenced by:  univ  4572