Theorem vpwex 4208

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 4209 from vpwex 4208. (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 3603	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4205	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4150	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2302	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1616	. . . 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 2769	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2266	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by:  pwexg  4209  pwnex  4480  exmidpw2en  6968  istopon  14181  dmtopon  14191  tgdom  14240