Theorem vpwex 4111

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 4112 from vpwex 4111. (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 3517	. 2 |- ~P x = { y | y C_ x }
2		axpow2 4108	. . . . 5 |- E. z A. y ( y C_ x -> y e. z )
3	2	bm1.3ii 4057	. . . 4 |- E. z A. y ( y e. z <-> y C_ x )
4		abeq2 2249	. . . . 5 |- ( z = { y | y C_ x } <-> A. y ( y e. z <-> y C_ x ) )
5	4	exbii 1585	. . . 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 2698	. 2 |- { y | y C_ x } e. _V
8	1, 7	eqeltri 2213	1 |- ~P x e. _V

Syntax hints:   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   {cab 2126   _Vcvv 2689   C_ wss 3076   ~Pcpw 3515

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by:  pwexg  4112  pwnex  4378  istopon  12219  dmtopon  12229  tgdom  12280