Theorem pwex 3932

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 3362	. . 3 |- ( z = A -> ~P z = ~P A )
3	2	eleq1d 2106	. 2 |- ( z = A -> ( ~P z e. _V <-> ~P A e. _V ) )
4		df-pw 3361	. . 3 |- ~P z = { y | y C_ z }
5		axpow2 3929	. . . . . 6 |- E. x A. y ( y C_ z -> y e. x )
6	5	bm1.3ii 3878	. . . . 5 |- E. x A. y ( y e. x <-> y C_ z )
7		abeq2 2146	. . . . . 6 |- ( x = { y | y C_ z } <-> A. y ( y e. x <-> y C_ z ) )
8	7	exbii 1496	. . . . 5 |- ( E. x x = { y | y C_ z } <-> E. x A. y ( y e. x <-> y C_ z ) )
9	6, 8	mpbir 134	. . . 4 |- E. x x = { y | y C_ z }
10	9	issetri 2564	. . 3 |- { y | y C_ z } e. _V
11	4, 10	eqeltri 2110	. 2 |- ~P z e. _V
12	1, 3, 11	vtocl 2608	1 |- ~P A e. _V

Syntax hints:   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  pwexg  3933  p0ex  3939  pp0ex  3940  ord3ex  3941  abexssex  5752  npex  6571  axcnex  6935  pnfxr  8692  mnfxr  8694  ixxex  8768