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Theorem vpwex 4006
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 4007 from vpwex 4006. (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 3427 . 2  |-  ~P x  =  { y  |  y 
C_  x }
2 axpow2 4003 . . . . 5  |-  E. z A. y ( y  C_  x  ->  y  e.  z )
32bm1.3ii 3952 . . . 4  |-  E. z A. y ( y  e.  z  <->  y  C_  x
)
4 abeq2 2196 . . . . 5  |-  ( z  =  { y  |  y  C_  x }  <->  A. y ( y  e.  z  <->  y  C_  x
) )
54exbii 1541 . . . 4  |-  ( E. z  z  =  {
y  |  y  C_  x }  <->  E. z A. y
( y  e.  z  <-> 
y  C_  x )
)
63, 5mpbir 144 . . 3  |-  E. z 
z  =  { y  |  y  C_  x }
76issetri 2628 . 2  |-  { y  |  y  C_  x }  e.  _V
81, 7eqeltri 2160 1  |-  ~P x  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  pwexg  4007
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