Theorem vpwex 4224

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 4225 from vpwex 4224. (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 3618	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4221	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4166	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2314	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1628	. . . 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 2781	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2278	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by:  pwexg  4225  pwnex  4497  exmidpw2en  7011  metuex  14350  istopon  14518  dmtopon  14528  tgdom  14577