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Theorem dmsn0 5473
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 5026 . 2  |-  -.  (/)  e.  ( _V  X.  _V )
2 dmsnn0 5471 . . 3  |-  ( (/)  e.  ( _V  X.  _V ) 
<->  dom  { (/) }  =/=  (/) )
32necon2bbii 2734 . 2  |-  ( dom 
{ (/) }  =  (/)  <->  -.  (/) 
e.  ( _V  X.  _V ) )
41, 3mpbir 209 1  |-  dom  { (/)
}  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {csn 4027    X. cxp 4997   dom cdm 4999
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-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  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-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009
This theorem is referenced by:  cnvsn0  5474  dmsnopss  5478  1st0  6787  2nd0  6788
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