ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwex Structured version   Unicode version

Theorem pwex 3923
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  _V
Assertion
Ref Expression
pwex  ~P  _V

Proof of Theorem pwex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfpowcl.1 . 2  _V
2 pweq 3354 . . 3  ~P  ~P
32eleq1d 2103 . 2  ~P  _V  ~P  _V
4 df-pw 3353 . . 3  ~P  {  | 
C_  }
5 axpow2 3920 . . . . . 6  C_
65bm1.3ii 3869 . . . . 5  C_
7 abeq2 2143 . . . . . 6  {  |  C_  }  C_
87exbii 1493 . . . . 5  {  |  C_  }  C_
96, 8mpbir 134 . . . 4  {  |  C_  }
109issetri 2558 . . 3  {  |  C_  }  _V
114, 10eqeltri 2107 . 2  ~P  _V
121, 3, 11vtocl 2602 1  ~P  _V
Colors of variables: wff set class
Syntax hints:   wb 98  wal 1240   wceq 1242  wex 1378   wcel 1390   {cab 2023   _Vcvv 2551    C_ wss 2911   ~Pcpw 3351
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by:  pwexg  3924  p0ex  3930  pp0ex  3931  ord3ex  3932  abexssex  5694  npex  6456  axcnex  6745  pnfxr  8462  mnfxr  8464  ixxex  8538
  Copyright terms: Public domain W3C validator