Theorem pwv 3735

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 3119	. . . 4 |- x C_ _V
2		vex 2689	. . . . 5 |- x e. _V
3	2	elpw 3516	. . . 4 |- ( x e. ~P _V <-> x C_ _V )
4	1, 3	mpbir 145	. . 3 |- x e. ~P _V
5	4, 2	2th 173	. 2 |- ( x e. ~P _V <-> x e. _V )
6	5	eqriv 2136	1 |- ~P _V = _V

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   ~Pcpw 3510

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by:  univ  4397