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Theorem pp0ex 4585
Description: The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
Assertion
Ref Expression
pp0ex  |-  { (/) ,  { (/) } }  e.  _V

Proof of Theorem pp0ex
StepHypRef Expression
1 pwpw0 4122 . 2  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
2 p0ex 4583 . . 3  |-  { (/) }  e.  _V
32pwex 4579 . 2  |-  ~P { (/)
}  e.  _V
41, 3eqeltrri 2489 1  |-  { (/) ,  { (/) } }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1844   _Vcvv 3061   (/)c0 3740   ~Pcpw 3957   {csn 3974   {cpr 3976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-pw 3959  df-sn 3975  df-pr 3977
This theorem is referenced by:  ord3ex  4586  zfpair  4630
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