Theorem vpwex 4158

Description: Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4159 from vpwex 4158. (Revised by BJ, 10-Aug-2022.)

Assertion

Ref	Expression
vpwex	|- ~P x e. _V

Proof of Theorem vpwex

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pw 3561	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4155	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4103	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2275	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1593	. . . 4 |- ( E. z z = { y | y C_ x } <-> E. z A. y ( y e. z <-> y C_ x ) )
6	3, 5	mpbir 145	. . 3 |- E. z z = { y | y C_ x }
7	6	issetri 2735	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2239	1 |- ~P x e. _V

Syntax hints:   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   _Vcvv 2726   C_ wss 3116   ~Pcpw 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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

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

This theorem is referenced by:  pwexg  4159  pwnex  4427  istopon  12651  dmtopon  12661  tgdom  12712