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Theorem pwexb 4172
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexb  _V  ~P  _V

Proof of Theorem pwexb
StepHypRef Expression
1 uniexb 4171 . 2  ~P  _V  U. ~P  _V
2 unipw 3944 . . 3  U. ~P
32eleq1i 2100 . 2  U. ~P  _V  _V
41, 3bitr2i 174 1  _V  ~P  _V
Colors of variables: wff set class
Syntax hints:   wb 98   wcel 1390   _Vcvv 2551   ~Pcpw 3351   U.cuni 3571
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-un 4136
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-nfc 2164  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-uni 3572
This theorem is referenced by: (None)
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