Theorem vpwex 4213

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 4214 from vpwex 4213. (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 3608	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4210	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4155	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2305	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1619	. . . 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 2772	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2269	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   _Vcvv 2763   C_ wss 3157   ~Pcpw 3606

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765  df-in 3163  df-ss 3170  df-pw 3608

This theorem is referenced by:  pwexg  4214  pwnex  4485  exmidpw2en  6982  metuex  14187  istopon  14333  dmtopon  14343  tgdom  14392