Theorem vpwex 4311

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 4312 from vpwex 4311. (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 3687	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4308	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4249	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2347	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1658	. . . 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 2831	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2311	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  pwexg  4312  pwnex  4590  exmidpw2en  7209  metuex  14864  istopon  15037  dmtopon  15047  tgdom  15096