Theorem vpwex 4263

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 4264 from vpwex 4263. (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 3651	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4260	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4205	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2338	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1651	. . . 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 2809	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2302	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  pwexg  4264  pwnex  4540  exmidpw2en  7074  metuex  14519  istopon  14687  dmtopon  14697  tgdom  14746