Theorem pwv 3834

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 3201	. . . 4 |- x C_ _V
2		vex 2763	. . . . 5 |- x e. _V
3	2	elpw 3607	. . . 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 2190	1 |- ~P _V = _V

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by:  univ  4507