MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  p0ex Structured version   Unicode version

Theorem p0ex 4634
Description: The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4635. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
p0ex  |-  { (/) }  e.  _V

Proof of Theorem p0ex
StepHypRef Expression
1 pw0 4174 . 2  |-  ~P (/)  =  { (/)
}
2 0ex 4577 . . 3  |-  (/)  e.  _V
32pwex 4630 . 2  |-  ~P (/)  e.  _V
41, 3eqeltrri 2552 1  |-  { (/) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ~Pcpw 4010   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028
This theorem is referenced by:  pp0ex  4636  dtruALT  4679  zfpair  4684  snsn0non  4996  opthprc  5046  fvclex  6753  tposexg  6966  2dom  7585  map1  7591  endisj  7601  pw2eng  7620  dfac4  8499  dfac2  8507  cdaval  8546  axcc2lem  8812  axdc2lem  8824  axcclem  8833  axpowndlem3  8971  axpowndlem3OLD  8972  ccatfn  12552  isstruct2  14495  plusffval  15740  staffval  17279  scaffval  17313  lpival  17675  ipffval  18450  tgdif0  19260  filcon  20119  alexsubALTlem2  20283  nmfval  20844  tchex  21395  tchnmfval  21406  legval  23698  oms0  27906  rankeq1o  29405  ssoninhaus  29490  onint1  29491  rrnval  29926  bnj105  32857  bj-tagex  33626  bj-1uplex  33647  lsatset  33787
  Copyright terms: Public domain W3C validator