Theorem vpwex 4269

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 4270 from vpwex 4269. (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 3654	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4266	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4210	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2340	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1653	. . . 4 |- ( E. z z = { y | y C_ x } <-> E. z A. y ( y e. z <-> y C_ x ) )
6	3, 5	mpbir 146	. . 3 |- E. z z = { y | y C_ x }
7	6	issetri 2812	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2304	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1395   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652

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-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  pwexg  4270  pwnex  4546  exmidpw2en  7103  metuex  14568  istopon  14736  dmtopon  14746  tgdom  14795