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Theorem dmsn0el 5292
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 5288 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 0nelelxp 4851 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  -.  (/)  e.  A
)
31, 2sylbir 213 . 2  |-  ( dom 
{ A }  =/=  (/) 
->  -.  (/)  e.  A )
43necon4ai 2641 1  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058   (/)c0 3737   {csn 3971    X. cxp 4820   dom cdm 4822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-dm 4832
This theorem is referenced by:  dmsnsnsn  5301
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