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Theorem dmsn0 5304
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 4873 . 2  |-  -.  (/)  e.  ( _V  X.  _V )
2 dmsnn0 5302 . . 3  |-  ( (/)  e.  ( _V  X.  _V ) 
<->  dom  { (/) }  =/=  (/) )
32necon2bbii 2631 . 2  |-  ( dom 
{ (/) }  =  (/)  <->  -.  (/) 
e.  ( _V  X.  _V ) )
41, 3mpbir 201 1  |-  dom  { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2924   (/)c0 3596   {csn 3782    X. cxp 4843   dom cdm 4845
This theorem is referenced by:  cnvsn0  5305  dmsnopss  5309  1st0  6320  2nd0  6321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-dm 4855
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