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Theorem dmsn0el 5142
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 5138 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 0nelelxp 4718 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  -.  (/)  e.  A
)
31, 2sylbir 204 . 2  |-  ( dom 
{ A }  =/=  (/) 
->  -.  (/)  e.  A )
43necon4ai 2505 1  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640    X. cxp 4687   dom cdm 4689
This theorem is referenced by:  dmsnsnsn  5151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699
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