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Theorem pp0ex 4390
Description: The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
pp0ex  |-  { (/) ,  { (/) } }  e.  _V

Proof of Theorem pp0ex
StepHypRef Expression
1 pwpw0 3948 . 2  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
2 p0ex 4388 . . 3  |-  { (/) }  e.  _V
32pwex 4384 . 2  |-  ~P { (/)
}  e.  _V
41, 3eqeltrri 2509 1  |-  { (/) ,  { (/) } }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1726   _Vcvv 2958   (/)c0 3630   ~Pcpw 3801   {csn 3816   {cpr 3817
This theorem is referenced by:  ord3ex  4391  zfpair  4403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823
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