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Theorem pp0ex 4631
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 4170 . 2  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
2 p0ex 4629 . . 3  |-  { (/) }  e.  _V
32pwex 4625 . 2  |-  ~P { (/)
}  e.  _V
41, 3eqeltrri 2547 1  |-  { (/) ,  { (/) } }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1762   _Vcvv 3108   (/)c0 3780   ~Pcpw 4005   {csn 4022   {cpr 4024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-pw 4007  df-sn 4023  df-pr 4025
This theorem is referenced by:  ord3ex  4632  zfpair  4679
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