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Theorem dmsn0el 5744
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 (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Proof of Theorem dmsn0el
StepHypRef Expression
1 dmsnn0 5740 . . 3 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2 0nelelxp 5284 . . 3 (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
31, 2sylbir 225 . 2 (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
43necon4ai 2974 1 (∅ ∈ 𝐴 → dom {𝐴} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  c0 4063  {csn 4317   × cxp 5248  dom cdm 5250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-dm 5260
This theorem is referenced by:  dmsnsnsn  5754
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