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Theorem pwex 3960
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.
StepHypRef Expression
1 zfpowcl.1 . 2  |-  A  e. 
_V
2 pweq 3390 . . 3  |-  ( z  =  A  ->  ~P z  =  ~P A
)
32eleq1d 2122 . 2  |-  ( z  =  A  ->  ( ~P z  e.  _V  <->  ~P A  e.  _V )
)
4 df-pw 3389 . . 3  |-  ~P z  =  { y  |  y 
C_  z }
5 axpow2 3957 . . . . . 6  |-  E. x A. y ( y  C_  z  ->  y  e.  x
)
65bm1.3ii 3906 . . . . 5  |-  E. x A. y ( y  e.  x  <->  y  C_  z
)
7 abeq2 2162 . . . . . 6  |-  ( x  =  { y  |  y  C_  z }  <->  A. y ( y  e.  x  <->  y  C_  z
) )
87exbii 1512 . . . . 5  |-  ( E. x  x  =  {
y  |  y  C_  z }  <->  E. x A. y
( y  e.  x  <->  y 
C_  z ) )
96, 8mpbir 138 . . . 4  |-  E. x  x  =  { y  |  y  C_  z }
109issetri 2581 . . 3  |-  { y  |  y  C_  z }  e.  _V
114, 10eqeltri 2126 . 2  |-  ~P z  e.  _V
121, 3, 11vtocl 2625 1  |-  ~P A  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389
This theorem is referenced by:  pwexg  3961  p0ex  3967  pp0ex  3968  ord3ex  3969  abexssex  5780  npex  6629  axcnex  6993  pnfxr  8793  mnfxr  8795  ixxex  8869
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