Theorem pwex 3953

Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Hypothesis

Ref	Expression
zfpowcl.1	|- A e. _V

Assertion

Ref	Expression
pwex	|- ~P A e. _V

Proof of Theorem pwex

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfpowcl.1	. 2 |- A e. _V
2		pweq 3385	. . 3 |- ( z = A -> ~P z = ~P A )
3	2	eleq1d 2147	. 2 |- ( z = A -> ( ~P z e. _V <-> ~P A e. _V ) )
4		df-pw 3384	. . 3 |- ~P z = { y | y C_ z }
5		axpow2 3950	. . . . . 6 |- E. x A. y ( y C_ z -> y e. x )
6	5	bm1.3ii 3899	. . . . 5 |- E. x A. y ( y e. x <-> y C_ z )
7		abeq2 2187	. . . . . 6 |- ( x = { y | y C_ z } <-> A. y ( y e. x <-> y C_ z ) )
8	7	exbii 1536	. . . . 5 |- ( E. x x = { y | y C_ z } <-> E. x A. y ( y e. x <-> y C_ z ) )
9	6, 8	mpbir 144	. . . 4 |- E. x x = { y | y C_ z }
10	9	issetri 2608	. . 3 |- { y | y C_ z } e. _V
11	4, 10	eqeltri 2151	. 2 |- ~P z e. _V
12	1, 3, 11	vtocl 2653	1 |- ~P A e. _V

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  pwexg  3954  p0ex  3959  pp0ex  3960  ord3ex  3961  abexssex  5772  npex  6663  axcnex  7027  pnfxr  8846  mnfxr  8848  ixxex  8922