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Theorem pp0ex 4215
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 3779 . 2  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
2 p0ex 4213 . . 3  |-  { (/) }  e.  _V
32pwex 4209 . 2  |-  ~P { (/)
}  e.  _V
41, 3eqeltrri 2367 1  |-  { (/) ,  { (/) } }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638   {csn 3653   {cpr 3654
This theorem is referenced by:  ord3ex  4216  zfpair  4228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660
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