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Theorem vpwex 4200
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 4201 from vpwex 4200. (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.
StepHypRef Expression
1 df-pw 3595 . 2  |-  ~P x  =  { y  |  y 
C_  x }
2 axpow2 4197 . . . . 5  |-  E. z A. y ( y  C_  x  ->  y  e.  z )
32bm1.3ii 4142 . . . 4  |-  E. z A. y ( y  e.  z  <->  y  C_  x
)
4 abeq2 2298 . . . . 5  |-  ( z  =  { y  |  y  C_  x }  <->  A. y ( y  e.  z  <->  y  C_  x
) )
54exbii 1616 . . . 4  |-  ( E. z  z  =  {
y  |  y  C_  x }  <->  E. z A. y
( y  e.  z  <-> 
y  C_  x )
)
63, 5mpbir 146 . . 3  |-  E. z 
z  =  { y  |  y  C_  x }
76issetri 2761 . 2  |-  { y  |  y  C_  x }  e.  _V
81, 7eqeltri 2262 1  |-  ~P x  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160   {cab 2175   _Vcvv 2752    C_ wss 3144   ~Pcpw 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754  df-in 3150  df-ss 3157  df-pw 3595
This theorem is referenced by:  pwexg  4201  pwnex  4470  exmidpw2en  6944  istopon  13998  dmtopon  14008  tgdom  14057
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