MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0elpw Unicode version

Theorem 0elpw 4180
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 3483 . 2  |-  (/)  C_  A
2 0ex 4150 . . 3  |-  (/)  e.  _V
32elpw 3631 . 2  |-  ( (/)  e.  ~P A  <->  (/)  C_  A
)
41, 3mpbir 200 1  |-  (/)  e.  ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   (/)c0 3455   ~Pcpw 3625
This theorem is referenced by:  marypha1lem  7186  brwdom2  7287  canthwdom  7293  pwcdadom  7842  isfin1-3  8012  canthp1lem2  8275  ixxssxr  10668  incexc  12296  smupf  12669  hashbc0  13052  ramz2  13071  mreexexlem3d  13548  acsfn  13561  isdrs2  14073  fpwipodrs  14267  clsval2  16787  mretopd  16829  alexsubALTlem2  17742  alexsubALTlem4  17744  esum0  23428  esumcst  23436  esumpcvgval  23446  prsiga  23492  indf1ofs  23609  kur14  23747  eupath2  23904  0hf  24807  sallnei  25529  comppfsc  26307  0totbnd  26497  heiborlem6  26540  istopclsd  26775
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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
  Copyright terms: Public domain W3C validator