Theorem vpwex 4181

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 4182 from vpwex 4181. (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 3579	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4178	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4126	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2286	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1605	. . . 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 2748	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2250	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579

This theorem is referenced by:  pwexg  4182  pwnex  4451  istopon  13552  dmtopon  13562  tgdom  13611