MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsn0el Structured version   Visualization version   GIF version

Theorem dmsn0el 6039
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 6035 . . 3 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2 0nelelxp 5558 . . 3 (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
31, 2sylbir 238 . 2 (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
43necon4ai 3021 1 (∅ ∈ 𝐴 → dom {𝐴} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2112  wne 2990  Vcvv 3444  c0 4246  {csn 4528   × cxp 5521  dom cdm 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-dm 5533
This theorem is referenced by:  dmsnsnsn  6048
  Copyright terms: Public domain W3C validator