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Theorem dmsn0el 5312
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 5308 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 0nelelxp 4868 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  -.  (/)  e.  A
)
31, 2sylbir 218 . 2  |-  ( dom 
{ A }  =/=  (/) 
->  -.  (/)  e.  A )
43necon4ai 2674 1  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   {csn 3959    X. cxp 4837   dom cdm 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-dm 4849
This theorem is referenced by:  dmsnsnsn  5321
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