Theorem vpwex 4103

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 4104 from vpwex 4103. (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 3512	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4100	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4049	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2248	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1584	. . . 4 |- ( E. z z = { y | y C_ x } <-> E. z A. y ( y e. z <-> y C_ x ) )
6	3, 5	mpbir 145	. . 3 |- E. z z = { y | y C_ x }
7	6	issetri 2695	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2212	1 |- ~P x e. _V

Syntax hints:   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   _Vcvv 2686   C_ wss 3071   ~Pcpw 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by:  pwexg  4104  pwnex  4370  istopon  12180  dmtopon  12190  tgdom  12241