Theorem vpwex 4174

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 4175 from vpwex 4174. (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 3574	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4171	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4119	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2284	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1603	. . . 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 2744	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2248	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1490   e. wcel 2146   {cab 2161   _Vcvv 2735   C_ wss 3127   ~Pcpw 3572

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574

This theorem is referenced by:  pwexg  4175  pwnex  4443  istopon  13080  dmtopon  13090  tgdom  13141