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Theorem dmsn0el 5475
Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
dmsn0el  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )

Proof of Theorem dmsn0el
StepHypRef Expression
1 dmsnn0 5471 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 0nelelxp 5027 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  -.  (/)  e.  A
)
31, 2sylbir 213 . 2  |-  ( dom 
{ A }  =/=  (/) 
->  -.  (/)  e.  A )
43necon4ai 2705 1  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   _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:  dmsnsnsn  5484
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