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Theorem p0ex 4197
Description: The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4198. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
p0ex  |-  { (/) }  e.  _V

Proof of Theorem p0ex
StepHypRef Expression
1 pw0 3762 . 2  |-  ~P (/)  =  { (/)
}
2 0ex 4150 . . 3  |-  (/)  e.  _V
32pwex 4193 . 2  |-  ~P (/)  e.  _V
41, 3eqeltrri 2354 1  |-  { (/) }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   {csn 3640
This theorem is referenced by:  pp0ex  4199  dtruALT  4207  zfpair  4212  snsn0non  4511  opthprc  4736  fvclex  5761  2dom  6933  map1  6939  endisj  6949  pw2eng  6968  dfac4  7749  dfac2  7757  cdaval  7796  axcc2lem  8062  axdc2lem  8074  axcclem  8083  axpowndlem3  8221  ccatfn  11427  isstruct2  13157  plusffval  14379  staffval  15612  scaffval  15645  lpival  15997  ipffval  16552  tgdif0  16730  filcon  17578  alexsubALTlem2  17742  nmfval  18111  tchex  18649  tchnmfval  18659  rankeq1o  24801  ssoninhaus  24887  onint1  24888  rrnval  26551  bnj105  28750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646
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