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Theorem dmsn0 5340
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 4909 . 2  |-  -.  (/)  e.  ( _V  X.  _V )
2 dmsnn0 5338 . . 3  |-  ( (/)  e.  ( _V  X.  _V ) 
<->  dom  { (/) }  =/=  (/) )
32necon2bbii 2662 . 2  |-  ( dom 
{ (/) }  =  (/)  <->  -.  (/) 
e.  ( _V  X.  _V ) )
41, 3mpbir 202 1  |-  dom  { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {csn 3816    X. cxp 4879   dom cdm 4881
This theorem is referenced by:  cnvsn0  5341  dmsnopss  5345  1st0  6356  2nd0  6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-dm 4891
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