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Theorem dmsn0el 6167
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 6163 . . 3 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2 0nelelxp 5672 . . 3 (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
31, 2sylbir 234 . 2 (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
43necon4ai 2972 1 (∅ ∈ 𝐴 → dom {𝐴} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wne 2940  Vcvv 3447  c0 4286  {csn 4590   × cxp 5635  dom cdm 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-dm 5647
This theorem is referenced by:  dmsnsnsn  6176
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