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Theorem 0elpw 4196
Description: Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
Assertion
Ref Expression
0elpw  |-  (/)  e.  ~P A

Proof of Theorem 0elpw
StepHypRef Expression
1 0ss 3496 . 2  |-  (/)  C_  A
2 0ex 4166 . . 3  |-  (/)  e.  _V
32elpw 3644 . 2  |-  ( (/)  e.  ~P A  <->  (/)  C_  A
)
41, 3mpbir 200 1  |-  (/)  e.  ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1696    C_ wss 3165   (/)c0 3468   ~Pcpw 3638
This theorem is referenced by:  marypha1lem  7202  brwdom2  7303  canthwdom  7309  pwcdadom  7858  isfin1-3  8028  canthp1lem2  8291  ixxssxr  10684  incexc  12312  smupf  12685  hashbc0  13068  ramz2  13087  mreexexlem3d  13564  acsfn  13577  isdrs2  14089  fpwipodrs  14283  clsval2  16803  mretopd  16845  alexsubALTlem2  17758  alexsubALTlem4  17760  esum0  23443  esumcst  23451  esumpcvgval  23461  prsiga  23507  indf1ofs  23624  kur14  23762  eupath2  23919  0hf  24879  sallnei  25632  comppfsc  26410  0totbnd  26600  heiborlem6  26643  istopclsd  26878
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-in 3172  df-ss 3179  df-nul 3469  df-pw 3640
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