MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsn0 Unicode version

Theorem dmsn0 5222
Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
Assertion
Ref Expression
dmsn0  |-  dom  { (/)
}  =  (/)

Proof of Theorem dmsn0
StepHypRef Expression
1 0nelxp 4799 . 2  |-  -.  (/)  e.  ( _V  X.  _V )
2 dmsnn0 5220 . . 3  |-  ( (/)  e.  ( _V  X.  _V ) 
<->  dom  { (/) }  =/=  (/) )
32necon2bbii 2577 . 2  |-  ( dom 
{ (/) }  =  (/)  <->  -.  (/) 
e.  ( _V  X.  _V ) )
41, 3mpbir 200 1  |-  dom  { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   {csn 3716    X. cxp 4769   dom cdm 4771
This theorem is referenced by:  cnvsn0  5223  dmsnopss  5227  1st0  6213  2nd0  6214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-dm 4781
  Copyright terms: Public domain W3C validator