Theorem vpwex 4239

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 4240 from vpwex 4239. (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 3628	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4236	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4181	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2316	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1629	. . . 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 2786	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2280	1 |- ~P x e. _V

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  pwexg  4240  pwnex  4514  exmidpw2en  7035  metuex  14432  istopon  14600  dmtopon  14610  tgdom  14659