Theorem dmsn0el 6202

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

Step	Hyp	Ref	Expression
1		dmsnn0 6198	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		0nelelxp 5687	. . 3 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
3	1, 2	sylbir 238	. 2 ⊢ (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
4	3	necon4ai 2991	1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  Vcvv 3457  ∅c0 4288  {csn 4585   × cxp 5650  dom cdm 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-dm 5662

This theorem is referenced by:  dmsnsnsn  6211