Theorem vpwex 4006

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 4007 from vpwex 4006. (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 3427	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4003	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 3952	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2196	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1541	. . . 4 |- ( E. z z = { y | y C_ x } <-> E. z A. y ( y e. z <-> y C_ x ) )
6	3, 5	mpbir 144	. . 3 |- E. z z = { y | y C_ x }
7	6	issetri 2628	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2160	1 |- ~P x e. _V

Syntax hints:   <-> wb 103   A.wal 1287   = wceq 1289   E.wex 1426   e. wcel 1438   {cab 2074   _Vcvv 2619   C_ wss 2997   ~Pcpw 3425

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427

This theorem is referenced by:  pwexg  4007