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Theorem 0elpw 4303
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 3592 . 2  |-  (/)  C_  A
2 0ex 4273 . . 3  |-  (/)  e.  _V
32elpw 3741 . 2  |-  ( (/)  e.  ~P A  <->  (/)  C_  A
)
41, 3mpbir 201 1  |-  (/)  e.  ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1717    C_ wss 3256   (/)c0 3564   ~Pcpw 3735
This theorem is referenced by:  marypha1lem  7366  brwdom2  7467  canthwdom  7473  pwcdadom  8022  isfin1-3  8192  canthp1lem2  8454  ixxssxr  10853  incexc  12537  smupf  12910  hashbc0  13293  ramz2  13312  mreexexlem3d  13791  acsfn  13804  isdrs2  14316  fpwipodrs  14510  clsval2  17030  mretopd  17072  alexsubALTlem2  17993  alexsubALTlem4  17995  eupath2  21543  esum0  24233  esumcst  24244  esumpcvgval  24257  prsiga  24303  kur14  24674  0hf  25825  comppfsc  26071  0totbnd  26166  heiborlem6  26209  istopclsd  26438
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-in 3263  df-ss 3270  df-nul 3565  df-pw 3737
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