Theorem vpwex 4165

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 4166 from vpwex 4165. (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 3568	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4162	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4110	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2279	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1598	. . . 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 2739	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2243	1 |- ~P x e. _V

Syntax hints:   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  pwexg  4166  pwnex  4434  istopon  12805  dmtopon  12815  tgdom  12866