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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

Proof of Theorem dmsn0el
StepHypRef Expression
1 dmsnn0 5308 . . 3
2 0nelelxp 4868 . . 3
31, 2sylbir 218 . 2
43necon4ai 2674 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1452   wcel 1904   wne 2641  cvv 3031  c0 3722  csn 3959   cxp 4837   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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