Theorem vpwex 4297

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 4298 from vpwex 4297. (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 3676	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4294	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4236	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2343	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1654	. . . 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 2825	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2307	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676

This theorem is referenced by:  pwexg  4298  pwnex  4575  exmidpw2en  7185  metuex  14829  istopon  15004  dmtopon  15014  tgdom  15063