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Theorem p0ex 4346
Description: The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4347. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
p0ex  |-  { (/) }  e.  _V

Proof of Theorem p0ex
StepHypRef Expression
1 pw0 3905 . 2  |-  ~P (/)  =  { (/)
}
2 0ex 4299 . . 3  |-  (/)  e.  _V
32pwex 4342 . 2  |-  ~P (/)  e.  _V
41, 3eqeltrri 2475 1  |-  { (/) }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2916   (/)c0 3588   ~Pcpw 3759   {csn 3774
This theorem is referenced by:  pp0ex  4348  dtruALT  4356  zfpair  4361  snsn0non  4659  opthprc  4884  fvclex  5940  tposexg  6452  2dom  7138  map1  7144  endisj  7154  pw2eng  7173  dfac4  7959  dfac2  7967  cdaval  8006  axcc2lem  8272  axdc2lem  8284  axcclem  8293  axpowndlem3  8430  ccatfn  11696  isstruct2  13433  plusffval  14657  staffval  15890  scaffval  15923  lpival  16271  ipffval  16834  tgdif0  17012  filcon  17868  alexsubALTlem2  18032  nmfval  18589  tchex  19129  tchnmfval  19139  rankeq1o  26016  ssoninhaus  26102  onint1  26103  rrnval  26426  bnj105  28795  lsatset  29473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589  df-pw 3761  df-sn 3780
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