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Theorem vpwex 4103
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 4104 from vpwex 4103. (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 3512 . 2  |-  ~P x  =  { y  |  y 
C_  x }
2 axpow2 4100 . . . . 5  |-  E. z A. y ( y  C_  x  ->  y  e.  z )
32bm1.3ii 4049 . . . 4  |-  E. z A. y ( y  e.  z  <->  y  C_  x
)
4 abeq2 2248 . . . . 5  |-  ( z  =  { y  |  y  C_  x }  <->  A. y ( y  e.  z  <->  y  C_  x
) )
54exbii 1584 . . . 4  |-  ( E. z  z  =  {
y  |  y  C_  x }  <->  E. z A. y
( y  e.  z  <-> 
y  C_  x )
)
63, 5mpbir 145 . . 3  |-  E. z 
z  =  { y  |  y  C_  x }
76issetri 2695 . 2  |-  { y  |  y  C_  x }  e.  _V
81, 7eqeltri 2212 1  |-  ~P x  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2125   _Vcvv 2686    C_ wss 3071   ~Pcpw 3510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512
This theorem is referenced by:  pwexg  4104  pwnex  4370  istopon  12194  dmtopon  12204  tgdom  12255
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